BSc: Control Theory

Control Theory

• Course name: Control Theory
• Code discipline: [S20]
• Subject area: Sensors and actuators; Robotic control.

Short Description

This course covers the following concepts: Introduction to Linear Control, Stability of linear dynamical systems; Controller design; Sensing, observers, Adaptive control.

Prerequisites

Prerequisite subjects

• CSE204 - Geometry And Linear Algebra II: Semidefinite matrices, Eigenvalues, Eigendecomposition (weak prerequisite), matrix exponentials (weak prerequisite), SVD (weak prerequisite)
• CSE205 - Differential Equations

Course Topics

Course Sections and Topics

Section	Topics within the section
Introduction to Linear Control, Stability of linear dynamical systems	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.
Controller design.	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.
Sensing, observers, Adaptive control	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.

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

Resources, literature and reference materials

Open access resources

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

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: ${\displaystyle {\dot {x}}=Ax+Bu}$ and a cost function: a) ${\textstyle J=\int (x^{\top }Qx+u^{\top }Iu)dt}$ b) ${\textstyle J=\int (x^{\top }Ix+u^{\top }Ru)dt}$ Write Riccati eq. and find LQR gain analytically.
2. You have a linear system a) ${\textstyle {\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\end{bmatrix}}={\begin{bmatrix}1&10\\-3&4\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}$ b) ${\textstyle {\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\end{bmatrix}}={\begin{bmatrix}-2&1\\2&40\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}$ Prove whether or not it is stable.
3. You have a linear system a) ${\displaystyle {\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\end{bmatrix}}={\begin{bmatrix}1&10\\-3&4\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}+{\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}}$ b) ${\displaystyle {\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\end{bmatrix}}={\begin{bmatrix}-2&1\\2&40\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}+{\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}}$ Your controller is: a) ${\textstyle {\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}={\begin{bmatrix}100&1\\1&20\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}$ b) ${\textstyle {\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}={\begin{bmatrix}7&2\\2&5\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}$ Prove whether the control system is stable.
4. You have linear dynamics:

a) ${\textstyle 2{\ddot {q}}+3{\dot {q}}-5q=u}$ b) ${\textstyle 10{\ddot {q}}-7{\dot {q}}+10q=u}$ c) ${\textstyle 15{\ddot {q}}+17{\dot {q}}+11q=2u}$ d) ${\textstyle 20{\ddot {q}}-{\dot {q}}-2q=-u}$ If ${\textstyle u=0}$ , which are stable (a - d)? Find ${\textstyle u}$ that makes the dynamics stable. Write transfer functions for the cases ${\textstyle u=0}$ and ${\textstyle u=-100x}$ .

1. 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