Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lecture Notes on Control System Theory and Design

Tamer Basar, Sean Meyn, William R. Perkins

TL;DR

These notes provide a comprehensive graduate-level treatment of control system theory with a focus on state-space methods. They cover modeling via state-space representations, stability via Lyapunov theory, and foundational concepts of controllability and observability, including time-varying extensions and canonical realizations. The text also develops practical tools such as state transition matrices, matrix exponentials, the Cayley–Hamilton theorem, and Grammians, illustrated through nonlinear and linear examples (e.g., pendula, RLC circuits). The material emphasizes synthesis in the form of observers/compensators and controllers and links theory to computation and simulation for robust feedback design in engineering applications.

Abstract

This is a collection of the lecture notes of the three authors for a first-year graduate course on control system theory and design (ECE 515 , formerly ECE 415) at the ECE Department of the University of Illinois at Urbana-Champaign. This is a fundamental course on the modern theory of dynamical systems and their control, and builds on a first-level course in control that emphasizes frequency-domain methods (such as the course ECE 486 , formerly ECE 386, at UIUC ). The emphasis in this graduate course is on state space techniques, and it encompasses modeling , analysis (of structural properties of systems, such as stability, controllability, and observability), synthesis (of observers/compensators and controllers) subject to design specifications, and optimization . Accordingly, this set of lecture notes is organized in four parts, with each part dealing with one of the issues identified above. Concentration is on linear systems , with nonlinear systems covered only in some specific contexts, such as stability and dynamic optimization. Both continuous-time and discrete-time systems are covered, with the former, however, in much greater depth than the latter. The main objective of this course is to teach the student some fundamental principles within a solid conceptual framework, that will enable her/him to design feedback loops compatible with the information available on the "states" of the system to be controlled, and by taking into account considerations such as stability, performance, energy conservation, and even robustness. A second objective is to familiarize her/him with the available modern computational, simulation, and general software tools that facilitate the design of effective feedback loops

Lecture Notes on Control System Theory and Design

TL;DR

These notes provide a comprehensive graduate-level treatment of control system theory with a focus on state-space methods. They cover modeling via state-space representations, stability via Lyapunov theory, and foundational concepts of controllability and observability, including time-varying extensions and canonical realizations. The text also develops practical tools such as state transition matrices, matrix exponentials, the Cayley–Hamilton theorem, and Grammians, illustrated through nonlinear and linear examples (e.g., pendula, RLC circuits). The material emphasizes synthesis in the form of observers/compensators and controllers and links theory to computation and simulation for robust feedback design in engineering applications.

Abstract

This is a collection of the lecture notes of the three authors for a first-year graduate course on control system theory and design (ECE 515 , formerly ECE 415) at the ECE Department of the University of Illinois at Urbana-Champaign. This is a fundamental course on the modern theory of dynamical systems and their control, and builds on a first-level course in control that emphasizes frequency-domain methods (such as the course ECE 486 , formerly ECE 386, at UIUC ). The emphasis in this graduate course is on state space techniques, and it encompasses modeling , analysis (of structural properties of systems, such as stability, controllability, and observability), synthesis (of observers/compensators and controllers) subject to design specifications, and optimization . Accordingly, this set of lecture notes is organized in four parts, with each part dealing with one of the issues identified above. Concentration is on linear systems , with nonlinear systems covered only in some specific contexts, such as stability and dynamic optimization. Both continuous-time and discrete-time systems are covered, with the former, however, in much greater depth than the latter. The main objective of this course is to teach the student some fundamental principles within a solid conceptual framework, that will enable her/him to design feedback loops compatible with the information available on the "states" of the system to be controlled, and by taking into account considerations such as stability, performance, energy conservation, and even robustness. A second objective is to familiarize her/him with the available modern computational, simulation, and general software tools that facilitate the design of effective feedback loops

Paper Structure

This paper contains 88 sections, 38 theorems, 656 equations, 42 figures.

Table of Contents

  1. System Modeling and Analysis
  2. State Space Models
  3. An electromechanical system
  4. Linearization about an equilibrium state
  5. Linearization about a trajectory
  6. A two link inverted pendulum
  7. An electrical circuit
  8. Transfer Functions & State Space Models
  9. Exercises
  10. Vector Spaces
  11. Fields
  12. Vector Space
  13. Bases
  14. Change of basis
  15. Linear Operators
  16. ...and 73 more sections

Key Result

Theorem 2.1

In any $n$-dimensional vector space, any set of $n$ linearly independent vectors qualifies as a basis. $\sqcap$$\sqcup$=0

Figures (42)

  • Figure 1: Magnetically Suspended Ball
  • Figure 2: Trajectory of a nonlinear state space model in two dimensions: $\dot x = f(x,u)$
  • Figure 3: The Pendubot
  • Figure 4: Coordinate description of the pendubot: $\ell_1$ is the length of the first link, and $\ell_{c1},\ell_{c2}$ are the distances to the center of mass of the respective links. The variables $q_1, q_2$ are joint angles of the respective links.
  • Figure 5: There is a continuum of different equilibrium positions for the Pendubot corresponding to different constant torque inputs $\tau$.
  • ...and 37 more figures

Theorems & Definitions (59)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 3.1: Cayley-Hamilton Theorem
  • Example 3.5.1
  • Example 3.5.2
  • Theorem 3.2
  • Lemma 3.3
  • Example 4.1.1
  • ...and 49 more