M.petrishchev: Created page with "= Control Theory = * '''Course name:''' Control Theory * '''Course number:''' [S20] * '''Knowledge area:''' Sensors and actuators; Robo..."

2022-06-28T11:01:04Z

Created page with "= Control Theory = * '''Course name:''' Control Theory * '''Course number:''' [S20] * '''Knowledge area:''' Sensors and actuators; Robo..."

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= Control Theory =

* '''Course name:''' Control Theory
* '''Course number:''' [S20]
* '''Knowledge area:''' Sensors and actuators; Robotic control.

== Course Characteristics ==

=== Key concepts of the class ===

* Introduction to Linear Control, Stability of linear dynamical systems
* Controller design
* Sensing, observers, Adaptive control

=== What is the 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.

== Prerequisites ==
* [https://eduwiki.innopolis.university/index.php/BSc:AnalyticGeometryAndLinearAlgebraI CSE204 - Geometry And Linear Algebra II]: Semidefinite matrices, Eigenvalues, Eigendecomposition (weak prerequisite), matrix exponentials (weak prerequisite), SVD (weak prerequisite)
* [https://eduwiki.innopolis.university/index.php/BSc:DifferentialEquations CSE205 - Differential Equations]

== Course Objectives Based on Bloom’s Taxonomy ==

=== What should a student remember at the end of the course? ===

By the end of the course, the students should be able to outline:

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

=== What should a student be able to understand at the end of the course? ===

By the end of the course, the students should be able to understand:

* 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

=== What should a student be able to apply at the end of the course? ===

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.

=== Course evaluation ===

{|
|+ Course grade breakdown
!align="center"|
!align="center"|
!align="center"| '''Proposed points'''
|-
|align="center"| Labs/seminar classes
|align="center"| 20
|align="center"| 30
|-
|align="center"| Interim performance assessment
|align="center"| 30
|align="center"| 20
|-
|align="center"| Exams
|align="center"| 50
|align="center"| 50
|}

=== Grades range ===

{|
|+ Course grading range
!align="center"|
!align="center"|
!align="center"| '''Proposed range'''
|-
|align="center"| A. Excellent
|align="center"| 90-100
|align="center"| 85-100
|-
|align="center"| B. Good
|align="center"| 75-89
|align="center"| 70-84
|-
|align="center"| C. Satisfactory
|align="center"| 60-74
|align="center"| 50-69
|-
|align="center"| D. Poor
|align="center"| 0-59
|align="center"| 0-49
|}

=== Resources and reference material ===

Main textbook:

* Ogata, K., 1994. Solving control engineering problems with MATLAB. Englewood Cliffs, NJ: Prentice-Hall.

Other reference material:

* 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.

== Course Sections ==

The main sections of the course and approximate hour distribution between them is as follows:

{|
|+ Course Sections
!align="center"| '''Section'''
! '''Section Title'''
!align="center"| '''Teaching Hours'''
|-
|align="center"| 1
| Introduction to Linear Control, Stability of linear dynamical systems
|align="center"| 6
|-
|align="center"| 2
| Controller design
|align="center"| 6
|-
|align="center"| 3
| Sensing, observers, Adaptive control
|align="center"| 6
|}

=== Section 1 ===

==== Section title: ====

Introduction to Linear Control, Stability of linear dynamical systems

=== Topics covered in this section: ===

* Control, introduction. Examples.
* Single input single output (SISO) systems. Block diagrams.
* 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.

=== What forms of evaluation were used to test students’ performance in this section? ===

{|
!align="center"|
!align="center"| '''Yes/No'''
|-
|align="center"| Development of individual parts of software product code
|align="center"| 0
|-
|align="center"| Homework and group projects
|align="center"| 1
|-
|align="center"| Midterm evaluation
|align="center"| 0
|-
|align="center"| Testing (written or computer based)
|align="center"| 1
|-
|align="center"| Reports
|align="center"| 1
|-
|align="center"| Essays
|align="center"| 0
|-
|align="center"| Oral polls
|align="center"| 0
|-
|align="center"| Discussions
|align="center"| 0
|}

=== Typical questions for ongoing performance evaluation within this section ===

# What is a linear dynamical system?
# What is an LTI system?
# What is an LTV system?
# Provide examples of LTI systems.
# What is a MIMO system?

=== Typical questions for seminar classes (labs) within this section ===

# 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.

=== Test questions for final assessment in this section ===

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

=== Section 2 ===

==== Section title: ====

Controller design.

=== Topics covered in this section: ===

* Stabilizing control. Control error.
* Proportional control.
* PD control. Order of a system and order of the controller.
* PID control.
* P, PD and PID control for DC motor.
* 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.

=== What forms of evaluation were used to test students’ performance in this section? ===

{|
!align="center"|
!align="center"| '''Yes/No'''
|-
|align="center"| Development of individual parts of software product code
|align="center"| 0
|-
|align="center"| Homework and group projects
|align="center"| 1
|-
|align="center"| Midterm evaluation
|align="center"| 1
|-
|align="center"| Testing (written or computer based)
|align="center"| 0
|-
|align="center"| Reports
|align="center"| 1
|-
|align="center"| Essays
|align="center"| 0
|-
|align="center"| Oral polls
|align="center"| 0
|-
|align="center"| Discussions
|align="center"| 1
|}

=== Typical questions for ongoing performance evaluation within this section ===

# What is stability in the sense of Lyapunov?
# What is stabilizing control?
# What is trajectory tracking?
# Why the control for a state-space system does not include the derivative of the state variable in the feedback law?
# How can a PD controller for a second-order linear mechanical system can be re-written in the state-space form?
# Write a closed-loop dynamics for an LTI system with a proportional controller.
# Give stability conditions for an LTI system with a proportional controller.
# Provide an example of a LTV system with negative eigenvalues that is not stable.
# Write algebraic Riccati equation for a standard additive quadratic cost.
# Derive algebraic Riccati equation for a given additive quadratic cost.
# Derive differential Riccati equation for a standard additive quadratic cost.
# What is the meaning of the unknown variable in the Riccati equation? What are its property in case of LTI dynamics.
# What is a frequency response?
# What is a phase response?

=== Typical questions for seminar classes (labs) within this section ===

# Design control for an LTI system using pole placement.
# Design control for an LTI system using Riccati (LQR).
# Simulate an LTI system with LQR controller.

=== Test questions for final assessment in this section ===

<ol>
<li>You have a linear system: <math display="block">\dot x = Ax + Bu</math> and a cost function: a) <math display="inline">J = \int (x^{\top} Q x + u^{\top} I u) dt</math> b) <math display="inline">J = \int (x^{\top} I x + u^{\top} R u) dt</math> Write Riccati eq. and find LQR gain analytically.</li>
<li>You have a linear system a) <math display="inline">\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 display="inline">\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.</li>
<li>You have a linear system a) <math display="block">\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 display="block">\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 display="inline">\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 display="inline">\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.</li>
<li>You have linear dynamics:
<div class="center">

{|
| a) <math display="inline">2 \ddot q + 3 \dot q - 5 q = u</math>
| b) <math display="inline">10 \ddot q - 7 \dot q + 10 q = u</math>
|-
| c) <math display="inline">15 \ddot q + 17 \dot q + 11 q = 2u</math>
| d) <math display="inline">20 \ddot q - \dot q - 2 q = -u</math>
|}

</div>
<ol>
<li>If <math display="inline">u = 0</math>, which are stable (a - d)?</li>
<li>Find <math display="inline">u</math> that makes the dynamics stable.</li>
<li>Write transfer functions for the cases <math display="inline">u = 0</math> and <math display="inline">u = -100x</math>.</li></ol>
</li>
<li>What is the difference between exponential stability, asymptotic stability and optimality?</li></ol>

=== Section 3 ===

==== Section title: ====

Sensing, observers, Adaptive control

==== Topics covered in this section: ====

* Modelling digital sensors: quantization, discretization, lag.
* Modelling sensor noise. Gaussian noise. Additive models. Multiplicative models. Dynamic sensor models.
* Observability.
* Filters.
* State observers.
* Optimal state observer for linear systems.
* Linearization of nonlinear systems.
* Linearization along trajectory.
* 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.

=== What forms of evaluation were used to test students’ performance in this section? ===

{|
!align="center"|
!align="center"| '''Yes/No'''
|-
|align="center"| Development of individual parts of software product code
|align="center"| 0
|-
|align="center"| Homework and group projects
|align="center"| 1
|-
|align="center"| Midterm evaluation
|align="center"| 0
|-
|align="center"| Testing (written or computer based)
|align="center"| 0
|-
|align="center"| Reports
|align="center"| 1
|-
|align="center"| Essays
|align="center"| 0
|-
|align="center"| Oral polls
|align="center"| 0
|-
|align="center"| Discussions
|align="center"| 0
|}

=== Typical questions for ongoing performance evaluation within this section ===

# What are the sources of sensor noise?
# How can we combat the lack of sensory information?
# When it is possible to combat the lack of sensory information?
# How can we combat the sensory noise?
# What is an Observer?
# What is a filter?
# How is additive noise different from multiplicative noise?

==== Typical questions for seminar classes (labs) within this section ====

# Simulate an LTI system with proportional control and sensor noise.
# Design an observer for an LTI system with proportional control and lack of sensory information.

==== Test questions for final assessment in this section ====

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

BSc: Control Theory.previous version - Revision history

M.petrishchev: Created page with "= Control Theory = * '''Course name:''' Control Theory * '''Course number:''' [S20] * '''Knowledge area:''' Sensors and actuators; Robo..."