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= Control Theory =
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= Analytical Geometry & Linear Algebra – I =
  
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* '''Course name''': Analytical Geometry & Linear Algebra – I
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* '''Course name''': Control Theory
 
* '''Code discipline''':
 
* '''Code discipline''':
−
* '''Subject area''': ['fundamental principles of vector algebra,', 'concepts of basic geometry objects and their transformations in the plane and in the space']
 +
* '''Subject area''': ['Introduction to Linear Control, Stability of linear dynamical systems', 'Controller design', 'Sensing, observers, Adaptive control']
   
 
== Short Description ==
  
== Short Description ==
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! Section !! Topics within the section
  
! Section !! Topics within the section
 
|-
  
|-
  +
| Introduction to Linear Control, Stability of linear dynamical systems ||
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| Vector algebra ||
 
  +
# Control, introduction. Examples.
−
# Vector spaces
  
  +
# Single input single output (SISO) systems. Block diagrams.
−
# Basic operations on vectors (summation, multiplication by scalar, dot product)
  
  +
# From linear differential equations to state space models.
−
# Linear dependency and in-dependency of the vectors
  
  +
# DC motor as a linear system.
−
# Basis in vector spaces
  
  +
# 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. ||
−
| Introduction to matrices and determinants ||
 
  +
# Stabilizing control. Control error.
−
# Relationship between Linear Algebra and Analytical Geometry
  
  +
# Proportional control.
−
# Matrices 2x2, 3x3
  
  +
# PD control. Order of a system and order of the controller.
−
# Determinants 2x2, 3x3
  
  +
# PID control.
−
# Operations om matrices and determinants
  
  +
# P, PD and PID control for DC motor.
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# The rank of a matrix
  
  +
# Trajectory tracking. Control input types. Standard inputs (Heaviside step function, Dirac delta function, sine wave).
−
# Inverse matrix
  
  +
# Tuning PD and PID. Pole placement.
−
# Systems of linear equations
  
  +
# Formal statements about stability. Lyapunov theory.
−
# Changing basis and coordinates
  
  +
# 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 ||
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| Lines in the plane and in the space ||
 
  +
# Modelling digital sensors: quantization, discretization, lag.
−
# General equation of a line in the plane
  
  +
# Modelling sensor noise. Gaussian noise. Additive models. Multiplicative models. Dynamic sensor models.
−
# General parametric equation of a line in the space
  
  +
# Observability.
−
# Line as intersection between planes
  
  +
# Filters.
−
# Vector equation of a line
  
  +
# State observers.
−
# Distance from a point to a line
  
  +
# Optimal state observer for linear systems.
−
# Distance between lines
  
  +
# Linearization of nonlinear systems.
−
# Inter-positioning of lines
  
  +
# Linearization along trajectory.
−
|-
  
  +
# Linearization of Inverted pendulum dynamics.
−
| Planes in the space ||
 
  +
# Model errors. Differences between random disturbances and unmodeled dynamics/processes.
−
# General equation of a plane
  
  +
# Adaptive control.
−
# Normalized linear equation of a plane
  
−
# Vector equation of a plane
 +
# Control for sets of linear systems.
  +
# Discretization, discretization error.
−
# Parametric equation a plane
  
  +
# Control for discrete linear systems.
−
# Distance from a point to a plane
  
  +
# Stability of discrete linear systems.
−
# Projection of a vector on the plane
  
  +
|}
−
# Inter-positioning of lines and planes
  
−
# Cross Product of two vectors
  
−
# Triple Scalar Product
  
−
|-
  
−
| Quadratic curves ||
 
−
# Circle
  
−
# Ellipse
  
−
# Hyperbola
  
−
# Parabola
  
−
# Canonical equations
  
−
# Shifting of coordinate system
  
−
# Rotating of coordinate system
  
−
# Parametrization
  
−
|-
  
−
| Quadric surfaces ||
 
−
# General equation of the quadric surfaces
  
−
# Canonical equation of a sphere and ellipsoid
  
−
# Canonical equation of a hyperboloid and paraboloid
  
−
# Surfaces of revolution
  
−
# Canonical equation of a cone and cylinder
  
−
# Vector equations of some quadric surfaces
  
−
|}
  
 
== 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.
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This is an introductory course in analytical geometry and linear algebra. After having studied the course, students get to know fundamental principles of vector algebra and its applications in solving various geometry problems, different types of equations of lines and planes, conics and quadric surfaces, transformations in the plane and in the space. An introduction on matrices and determinants as a fundamental knowledge of linear algebra is also provided.
  
   
 
=== ILOs defined at three levels ===
  
=== ILOs defined at three levels ===
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==== 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)
−
* List basic notions of vector algebra,
  
  +
* methods for controller analysis
−
* recite the base form of the equations of transformations in planes and spaces,
  
  +
* methods for sensory data processing for linear systems
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* recall equations of lines and planes,
  
−
* identify the type of conic section,
  
−
* recognize the kind of quadric surfaces.
  
   
 
==== 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
−
* explain the geometrical interpretation of the basic operations of vector algebra,
  
  +
* Eigenvalue analysis for linear systems
−
* restate equations of lines and planes in different forms,
  
  +
* Proportional and PD controllers
−
* interpret the geometrical meaning of the conic sections in the mathematical expression,
  
  +
* How to stabilize a linear system
−
* give the examples of the surfaces of revolution,
  
  +
* Lyapunov Stability
−
* understand the value of geometry in various fields of science and techniques.
  
  +
* 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.
−
* Perform the basic operations of vector algebra,
  
  +
* Design a controller by solving Algebraic Riccati eq.
−
* use different types of equations of lines and planes to solve the plane and space problems,
  
  +
* Find if a system is stable or not, using eigenvalue analysis.
−
* represent the conic section in canonical form,
  
−
* compose the equation of quadric surface.
  
 
== Grading ==
  
== Grading ==
   
Line 113: Line 110:
 
! Grade !! Range !! Description of performance
  
! Grade !! Range !! Description of performance
 
|-
  
|-
−
| A. Excellent || 80-100 || -
 +
| A. Excellent || 85-100 || -
 
|-
  
|-
−
| B. Good || 60-79 || -
 +
| B. Good || 70-84 || -
 
|-
  
|-
−
| C. Satisfactory || 40-59 || -
 +
| C. Satisfactory || 50-69 || -
 
|-
  
|-
−
| D. Poor || 0-39 || -
 +
| D. Poor || 0-49 || -
 
|}
  
|}
   
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! Activity Type !! Percentage of the overall course grade
  
! Activity Type !! Percentage of the overall course grade
 
|-
  
|-
−
| Labs/seminar classes || 10
 +
| Labs/seminar classes || 30
 
|-
  
|-
 
| Interim performance assessment || 20
  
| Interim performance assessment || 20
 
|-
  
|-
−
| Exams || 70
 +
| Exams || 50
 
|}
  
|}
   
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=== Open access resources ===
  
=== 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 ===
  
=== Closed access resources ===
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=== Software and tools used within the course ===
  
=== Software and tools used within the course ===
  +
 
= Teaching Methodology: Methods, techniques, & activities =
  
= Teaching Methodology: Methods, techniques, & activities =
   
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|+ Activities within each section
  
|+ Activities within each section
 
|-
  
|-
−
! Learning Activities !! Section 1 !! Section 2 !! Section 3 !! Section 4 !! Section 5 !! Section 6
 +
! Learning Activities !! Section 1 !! Section 2 !! Section 3
 
|-
  
|-
−
| Homework and group projects || 1 || 1 || 1 || 1 || 1 || 1
 +
| Homework and group projects || 1 || 1 || 1
 
|-
  
|-
−
| Midterm evaluation || 1 || 1 || 1 || 1 || 1 || 1
 +
| Testing (written or computer based) || 1 || 0 || 0
 
|-
  
|-
−
| Testing (written or computer based) || 1 || 1 || 1 || 1 || 1 || 1
 +
| Reports || 1 || 1 || 1
 
|-
  
|-
−
| Discussions || 1 || 1 || 1 || 1 || 1 || 1
 +
| Midterm evaluation || 0 || 1 || 0
−
|}
 +
|-
  +
| Discussions || 0 || 1 || 0
  +
|}
 
== Formative Assessment and Course Activities ==
  
== Formative Assessment and Course Activities ==
   
Line 173: Line 174:
 
! Activity Type !! Content !! Is Graded?
  
! Activity Type !! Content !! Is Graded?
 
|-
  
|-
−
| Question || How to perform the shift of the vector? || 1
 +
| Question || What is a linear dynamical system? || 1
 
|-
  
|-
−
| Question || What is the geometrical interpretation of the dot product? || 1
 +
| Question || What is an LTI system? || 1
 
|-
  
|-
−
| Question || How to determine whether the vectors are linearly dependent? || 1
 +
| Question || What is an LTV system? || 1
 
|-
  
|-
−
| Question || What is a vector basis? || 1
 +
| Question || Provide examples of LTI systems || 1
 
|-
  
|-
  +
| Question || What is a MIMO system? || 1
−
| Question || Evaluate <math>{\textstyle |{\textbf {a}}|^{2}-2{\sqrt {3}}{\textbf {a}}\cdot {\textbf {b}}-7|{\textbf {b}}|^{2}}</math> given that <math>{\textstyle |{\textbf {a}}|=4}</math> , <math>{\textstyle |{\textbf {b}}|=1}</math> , <math>{\textstyle \angle ({\textbf {a}},\,{\textbf {b}})=150^{\circ }}</math> || 0
  
 
|-
  
|-
  +
| 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 || Prove that vectors <math>{\textstyle {\textbf {b}}({\textbf {a}}\cdot {\textbf {c}})-{\textbf {c}}({\textbf {a}}\cdot {\textbf {b}})}</math> and <math>{\textstyle {\textbf {a}}}</math> are perpendicular to each other || 0
  
−
|-
 +
|}
−
| Question || Bases <math>{\textstyle AD}</math> and <math>{\textstyle BC}</math> of trapezoid <math>{\textstyle ABCD}</math> are in the ratio of <math>{\textstyle 4:1}</math> The diagonals of the trapezoid intersect at point <math>{\textstyle M}</math> and the extensions of sides <math>{\textstyle AB}</math> and <math>{\textstyle CD}</math> intersect at point <math>{\textstyle P}</math> Let us consider the basis with <math>{\textstyle A}</math> as the origin, <math>{\textstyle {\overrightarrow {AD}}}</math> and <math>{\textstyle {\overrightarrow {AB}}}</math> as basis vectors Find the coordinates of points <math>{\textstyle M}</math> and <math>{\textstyle P}</math> in this basis || 0
  
−
|-
  
−
| Question || A line segment joining a vertex of a tetrahedron with the centroid of the opposite face (the centroid of a triangle is an intersection point of all its medians) is called a median of this tetrahedron Using vector algebra prove that all the four medians of any tetrahedron concur in a point that divides these medians in the ratio of <math>{\textstyle 3:1}</math> , the longer segments being on the side of the vertex of the tetrahedron || 0
  
−
|}
  
 
==== Section 2 ====
  
==== Section 2 ====
 
{| class="wikitable"
  
{| class="wikitable"
Line 195: Line 192:
 
! Activity Type !! Content !! Is Graded?
  
! Activity Type !! Content !! Is Graded?
 
|-
  
|-
−
| Question || What is the difference between matrices and determinants? || 1
 +
| Question || What is stability in the sense of Lyapunov? || 1
 
|-
  
|-
  +
| Question || What is stabilizing control? || 1
−
| Question || Matrices <math>{\textstyle A}</math> and <math>{\textstyle C}</math> have dimensions of <math>{\textstyle m\times n}</math> and <math>{\textstyle p\times q}</math> respectively, and it is known that the product <math>{\textstyle ABC}</math> exists What are possible dimensions of <math>{\textstyle B}</math> and <math>{\textstyle ABC}</math> ? || 1
  
 
|-
  
|-
−
| Question || How to determine the rank of a matrix? || 1
 +
| Question || What is trajectory tracking? || 1
 
|-
  
|-
−
| Question || What is the meaning of the inverse matrix? || 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 to restate a system of linear equations in the matrix form? || 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 || Find <math>{\textstyle A+B}</math> and <math>{\textstyle 2A-3B+I}</math> || 0
  
 
|-
  
|-
  +
| Question || Give stability conditions for an LTI system with a proportional controller || 1
−
| Question || Find the products <math>{\textstyle AB}</math> and <math>{\textstyle BA}</math> (and so make sure that, in general, <math>{\textstyle AB\neq BA}</math> for matrices) || 0
  
 
|-
  
|-
−
| Question || Find the inverse matrices for the given ones || 0
 +
| Question || Provide an example of a LTV system with negative eigenvalues that is not stable || 1
 
|-
  
|-
−
| Question || Find the determinants of the given matrices || 0
 +
| 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 || Point <math>{\textstyle M}</math> is the centroid of face <math>{\textstyle BCD}</math> of tetrahedron <math>{\textstyle ABCD}</math> The old coordinate system is given by <math>{\textstyle A}</math> , <math>{\textstyle {\overrightarrow {AB}}}</math> , <math>{\textstyle {\overrightarrow {AC}}}</math> , <math>{\textstyle {\overrightarrow {AD}}}</math> , and the new coordinate system is given by <math>{\textstyle M}</math> , <math>{\textstyle {\overrightarrow {MB}}}</math> , <math>{\textstyle {\overrightarrow {MC}}}</math> , <math>{\textstyle {\overrightarrow {MA}}}</math> Find the coordinates of a point in the old coordinate system given its coordinates <math>{\textstyle x'}</math> , <math>{\textstyle y'}</math> , <math>{\textstyle z'}</math> in the new one || 0
  
−
|}
  
−
==== Section 3 ====
  
−
{| class="wikitable"
  
−
|+
 
 
|-
  
|-
  +
| Question || Derive differential Riccati equation for a standard additive quadratic cost || 1
−
! Activity Type !! Content !! Is Graded?
  
 
|-
  
|-
−
| Question || How to represent a line in the vector form? || 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 the result of intersection of two planes in vector form? || 1
 +
| Question || What is a frequency response? || 1
 
|-
  
|-
−
| Question || How to derive the formula for the distance from a point to a line? || 1
 +
| Question || What is a phase response? || 1
 
|-
  
|-
−
| Question || How to interpret geometrically the distance between lines? || 1
 +
| Question || Design control for an LTI system using pole placement || 0
 
|-
  
|-
−
| Question || List all possible inter-positions of lines in the space || 1
 +
| Question || Design control for an LTI system using Riccati (LQR) || 0
 
|-
  
|-
  +
| Question || Simulate an LTI system with LQR controller || 0
−
| Question || Two lines are given by the equations <math>{\textstyle {\textbf {r}}\cdot {\textbf {n}}=A}</math> and <math>{\textstyle {\textbf {r}}={\textbf {r}}_{0}+{\textbf {a}}t}</math> , and at that <math>{\textstyle {\textbf {a}}\cdot {\textbf {n}}\neq 0}</math> Find the position vector of the intersection point of these lines || 0
  
−
|-
 +
|}
  +
==== Section 3 ====
−
| Question || Find the distance from point <math>{\textstyle M_{0}}</math> with the position vector <math>{\textstyle {\textbf {r}}_{0}}</math> to the line defined by the equation (a) <math>{\textstyle {\textbf {r}}={\textbf {r}}_{0}+{\textbf {a}}t}</math> ; (b) <math>{\textstyle {\textbf {r}}\cdot {\textbf {n}}=A}</math> || 0
  
−
|-
  
−
| Question || Diagonals of a rhombus intersect at point <math>{\textstyle M(1;\,2)}</math> , the longest of them being parallel to a horizontal axis The side of the rhombus equals 2 and its obtuse angle is <math>{\textstyle 120^{\circ }}</math> Compose the equations of the sides of this rhombus || 0
  
−
|-
  
−
| Question || Compose the equations of lines passing through point <math>{\textstyle A(2;-4)}</math> and forming angles of <math>{\textstyle 60^{\circ }}</math> with the line <math>{\textstyle {\frac {1-2x}{3}}={\frac {3+2y}{-2}}}</math> || 0
  
−
|}
  
−
==== Section 4 ====
  
 
{| class="wikitable"
  
{| class="wikitable"
 
|+
 
|+
Line 245: Line 232:
 
! Activity Type !! Content !! Is Graded?
  
! Activity Type !! Content !! Is Graded?
 
|-
  
|-
−
| Question || What is the difference between general and normalized forms of equations of a plane? || 1
 +
| Question || What are the sources of sensor noise? || 1
 
|-
  
|-
−
| Question || How to rewrite the equation of a plane in a vector form? || 1
 +
| Question || How can we combat the lack of sensory information? || 1
 
|-
  
|-
−
| Question || What is the normal to a plane? || 1
 +
| Question || When it is possible to combat the lack of sensory information? || 1
 
|-
  
|-
−
| Question || How to interpret the cross products of two vectors? || 1
 +
| Question || How can we combat the sensory noise? || 1
 
|-
  
|-
−
| Question || What is the meaning of scalar triple product of three vectors? || 1
 +
| Question || What is an Observer? || 1
 
|-
  
|-
  +
| Question || What is a filter? || 1
−
| Question || Find the cross product of (a) vectors <math>{\textstyle {\textbf {a}}(3;-2;\,1)}</math> and <math>{\textstyle {\textbf {b}}(2;-5;-3)}</math> ; (b) vectors <math>{\textstyle {\textbf {a}}(3;-2;\,1)}</math> and <math>{\textstyle {\textbf {c}}(-18;\,12;-6)}</math> || 0
  
 
|-
  
|-
  +
| Question || How is additive noise different from multiplicative noise? || 1
−
| Question || A triangle is constructed on vectors <math>{\textstyle {\textbf {a}}(2;4;-1)}</math> and <math>{\textstyle {\textbf {b}}(-2;1;1)}</math> (a) Find the area of this triangle (b) Find the altitudes of this triangle || 0
  
 
|-
  
|-
  +
| Question || Simulate an LTI system with proportional control and sensor noise || 0
−
| Question || Find the scalar triple product of <math>{\textstyle {\textbf {a}}(1;\,2;-1)}</math> , <math>{\textstyle {\textbf {b}}(7;3;-5)}</math> , <math>{\textstyle {\textbf {c}}(3;\,4;-3)}</math> || 0
  
 
|-
  
|-
  +
| Question || Design an observer for an LTI system with proportional control and lack of sensory information || 0
−
| Question || It is known that basis vectors <math>{\textstyle {\textbf {e}}_{1}}</math> , <math>{\textstyle {\textbf {e}}_{2}}</math> , <math>{\textstyle {\textbf {e}}_{3}}</math> have lengths of <math>{\textstyle 1}</math> , <math>{\textstyle 2}</math> , <math>{\textstyle 2{\sqrt {2}}}</math> respectively, and <math>{\textstyle \angle ({\textbf {e}}_{1},{\textbf {e}}_{2})=120^{\circ }}</math> , <math>{\textstyle \angle ({\textbf {e}}_{1},{\textbf {e}}_{3})=135^{\circ }}</math> , <math>{\textstyle \angle ({\textbf {e}}_{2},{\textbf {e}}_{3})=45^{\circ }}</math> Find the volume of a parallelepiped constructed on vectors with coordinates <math>{\textstyle (-1;\,0;\,2)}</math> , <math>{\textstyle (1;\,1\,4)}</math> and <math>{\textstyle (-2;\,1;\,1)}</math> in this basis || 0
  
−
|}
 +
|}
−
==== Section 5 ====
  
−
{| class="wikitable"
  
−
|+
 
−
|-
  
−
! Activity Type !! Content !! Is Graded?
  
−
|-
  
−
| Question || Formulate the canonical equation of the given quadratic curve || 1
  
−
|-
  
−
| Question || Which orthogonal transformations of coordinates do you know? || 1
  
−
|-
  
−
| Question || How to perform a transformation of the coordinate system? || 1
  
−
|-
  
−
| Question || How to represent a curve in the space? || 1
  
−
|-
  
−
| Question || Prove that a curve given by <math>{\textstyle 34x^{2}+24xy+41y^{2}-44x+58y+1=0}</math> is an ellipse Find the major and minor axes of this ellipse, its eccentricity, coordinates of its center and foci Find the equations of axes and directrices of this ellipse || 0
  
−
|-
  
−
| Question || Determine types of curves given by the following equations For each of the curves, find its canonical coordinate system (ie indicate the coordinates of origin and new basis vectors in the initial coordinate system) and its canonical equation (a) <math>{\textstyle 9x^{2}-16y^{2}-6x+8y-144=0}</math> ; (b) <math>{\textstyle 9x^{2}+4y^{2}+6x-4y-2=0}</math> ; (c) <math>{\textstyle 12x^{2}-12x-32y-29=0}</math> ; (d) <math>{\textstyle xy+2x+y=0}</math> ; || 0
  
−
|-
  
−
| Question || Find the equations of lines tangent to curve <math>{\textstyle 6xy+8y^{2}-12x-26y+11=0}</math> that are (a) parallel to line <math>{\textstyle 6x+17y-4=0}</math> ; (b) perpendicular to line <math>{\textstyle 41x-24y+3=0}</math> ; (c) parallel to line <math>{\textstyle y=2}</math> || 0
  
−
|}
  
−
==== Section 6 ====
  
−
{| class="wikitable"
  
−
|+
 
−
|-
  
−
! Activity Type !! Content !! Is Graded?
  
−
|-
  
−
| Question || What is the type of a quadric surface given by a certain equation? || 1
  
−
|-
  
−
| Question || How to compose the equation of a surface of revolution? || 1
  
−
|-
  
−
| Question || What is the difference between a directrix and generatrix? || 1
  
−
|-
  
−
| Question || How to represent a quadric surface in the vector form? || 1
  
−
|-
  
−
| Question || For each value of parameter <math>{\textstyle a}</math> determine types of surfaces given by the equations: (a) <math>{\textstyle x^{2}+y^{2}-z^{2}=a}</math> ; (b) <math>{\textstyle x^{2}+a\left(y^{2}+z^{2}\right)=1}</math> ; (c) <math>{\textstyle x^{2}+ay^{2}=az}</math> ; (d) <math>{\textstyle x^{2}+ay^{2}=az+1}</math> || 0
  
−
|-
  
−
| Question || Find a vector equation of a right circular cone with apex <math>{\textstyle M_{0}\left({\textbf {r}}_{0}\right)}</math> and axis <math>{\textstyle {\textbf {r}}={\textbf {r}}_{0}+{\textbf {a}}t}</math> if it is known that generatrices of this cone form the angle of <math>{\textstyle \alpha }</math> with its axis || 0
  
−
|-
  
−
| Question || Find the equation of a cylinder with radius <math>{\textstyle {\sqrt {2}}}</math> that has an axis <math>{\textstyle x=1+t}</math> , <math>{\textstyle y=2+t}</math> , <math>{\textstyle z=3+t}</math> || 0
  
−
|-
  
−
| Question || An ellipsoid is symmetric with respect to coordinate planes, passes through point <math>{\textstyle M(3;\,1;\,1)}</math> and circle <math>{\textstyle x^{2}+y^{2}+z^{2}=9}</math> , <math>{\textstyle x-z=0}</math> Find the equation of this ellipsoid || 0
  
−
|}
  
 
=== Final assessment ===
  
=== Final assessment ===
 
'''Section 1'''
  
'''Section 1'''
  +
# Convert a linear differential equation into a state space form
−
# Vector spaces General concepts
  
  +
# Convert a transfer function into a state space form
−
# Dot product as an operation on vectors
  
  +
# Convert a linear differential equation into a transfer function
−
# Basis in vector spaces Its properties
  
  +
# What does it mean for a linear differential equation to be stable?
 
'''Section 2'''
  
'''Section 2'''
  +
−
# Operations om matrices and determinants
  
−
# Inverse matrix
  
−
# Systems of linear equations and their solution in matrix form
  
−
# Changing basis and coordinates
  
 
'''Section 3'''
  
'''Section 3'''
  +
# Write a model of a linear system with additive Gaussian noise
−
# Lines in the plane and in the space Equations of lines
  
  +
# Derive and implement an observer
−
# Distance from a point to a line
  
  +
# Derive and implement a filter
−
# Distance between two parallel lines
  
−
# Distance between two skew lines
  
−
'''Section 4'''
  
−
# Planes in the space Equations of planes
  
−
# Distance from a point to a plane, from a line to a plane
  
−
# Projection of a vector on the plane
  
−
# Cross product, its properties and geometrical interpretation
  
−
# Scalar triple product, its properties and geometrical interpretation
  
−
'''Section 5'''
  
−
# Determine the type of a given curve with the use of the method of invariant
  
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# Compose the canonical equation of a given curve
  
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# Determine the canonical coordinate system for a given curve
  
−
'''Section 6'''
  
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# Determine the type of a quadric surface given by a certain equation
  
−
# Compose the equation of a surface of revolution with the given directrix and generatrix
  
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# Represent a given equation of a quadric surface in the vector form
  
   
 
=== The retake exam ===
  
=== The retake exam ===
Line 341: Line 269:
   
 
'''Section 3'''
  
'''Section 3'''
−  
−
'''Section 4'''
  
−  
−
'''Section 5'''
  
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−
'''Section 6'''
  

Revision as of 18:38, 19 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

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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