Difference between revisions of "IU:TestPage"

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# What does it mean for a linear differential equation to be stable?
  
# What does it mean for a linear differential equation to be stable?
 
'''Section 2'''
  
'''Section 2'''
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# <math>{\displaystyle {\dot {x}}=Ax+Bu}</math>
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# <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>
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# <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>
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# You have linear dynamics: <math>{\textstyle 2{\ddot {q}}+3{\dot {q}}-5q=u}</math> <math>{\textstyle u=0}</math>
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# <math>{\textstyle u=0}</math>
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# <math>{\textstyle u}</math>
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# <math>{\textstyle u=0}</math>
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# 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
  
# Write a model of a linear system with additive Gaussian noise

Revision as of 22:44, 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

  4. You have linear dynamics:
  8. 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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