ECE 515/ME 540: Fall 2024 Lecture Schedule

System modeling and analysis

Tue Aug 27
Ch. 1

Introduction and administrivia
State-space models
Linearization about an equilibrium point
Linearization about a trajectory

Thu Aug 29
Ch. 1

Input-output description of SISO LTI systems using transfer functions
State-space realization
Controllable, observable, modal canonical forms

Tue Sept 3
Ch. 2

Fields and vector spaces
Linear independence, bases, dimension
Change of basis
Linear operators and matrices

Thu Sept 5
Ch. 2

Linear operators: nullspace and range
Eigenvalues and eigenvectors
Diagonalization and Jordan canonical form

Tue Sep 10
Ch. 3

Solving state-space equations
State transition matrix
Matrix exponential
The Cayley-Hamilton theorem

Thu Sep 12
Ch. 3

Solving state-space equations: time-varying systems
The fundamental matrix and the state transition matrix
Peano-Baker series

System structural properties

Tue Sep 17
Ch. 4

Stability
Motivating example: external vs. internal stability, pole-zero cancellation
Stability in the sense of Lyapunov
Asymptotic and global asymptotic stability
Stability criteria for LTI systems
Lyapunov's direct method
LaSalle's invariance theorem

Thu Sep 19
Ch. 4

Stability (cont.)
Stability of linear time-invariant systems
Lyapunov equation
Nonlinear systems and linearization
Hartman-Grobman theorem (the easy part)
Input-output stability

Tue Sep 24
Ch. 5

Controllability
Motivation and definition
Controllability of linear time-invariant systems
Controllability matrix, rank criterion
The general LTV case: the controllability Grammian

Thu Sep 26

No class: Allerton conference

Tu Oct 1
Ch. 5

Controllability (cont.)
The general LTV case: the controllability Grammian (continued)
Invariance of controllability under change of coordinates
Other tests for controllability
Modal form, the Hautus-Rosenbrock criterion

Thu Oct 3
Exam 1 -- in class

Covers material through stability (i.e. problem sets and solutions 1-4, lectures 1-8, chapters 1-4 in notes).

Tue Oct 8
Ch. 5, 6

Observability, duality, minimality
Observability: motivation and definition
The observability matrix
The general LTV case: the observability Gramian

Thu Oct 10
Ch. 6

Observability (cont.)
Duality between controllability and observability
Kalman canonical forms
Realization of transfer functions
Minimal (controllable and observable) realization

Feedback

Tue Oct 15
Ch. 7

Pole placement
Kalman's canonical forms revisited
Stabilizability and closed-loop pole placement
Detectability and observer pole placement
Duality between controllability/observability and between stabilizability/detectability

Thu Oct 17
Ch. 7

Pole placement (cont.)
Dynamic output feedback
The separation principle
Reduced-order (Luenberger) observers

Tue Oct 22
Ch. 8.1

Ch. 8.1 Tracking and disturbance rejection
Internal model principle

Thu Oct 24
Ch. 8.1

Tracking and disturbance rejection (continued)
Controllability indices and Luenberger controllability canonical form for MIMO systems

Tue Oct 29
Ch. 8.2 and 9

Transfer function approach: Sylvester systems
IMP for linear time-invariant systems (see Section 1 of E.D. Sontag, "Adaptation and regulation with signal detection implies internal model")
Control goals: stability, regulation/tracking, transient response shaping, robustness
Sensitivity to plant model misspecification and disturbances
Fundamental limitations: Bode's sensitivity integral

Thu Oct 31

BIBO stability
Review and comment on stable subspace, controllable subspace, unobservable subspace, and Kalman cannonical forms.

Thu Nov 5

Review

Optimal Control

Thu Nov 7
Ch. 10.1

Begin optimal control Formulation of the finite-horizon optimal control problem
Bellman's dynamic programming principle for discrete time and discrete state

Tu Nov 12
Exam 2 -- in class

The exam will cover material through problem set 9 and Chapters 1-9, excluding reduced order observers, and lectures through November 5. Emphasis on material covered after exam 1 -- i.e. Chapters 5-9. You may bring two sheets of notes two sided to consult during the exam with font size 10 or lerger or equivalent handwriting size.

Thu Nov 14
Ch. 10.1-3

Dynamic progamming
Formulation of the finite-horizon optimal control problem
Bellman's dynamic programming principle
The Hamilton-Jacobi-Bellman equation
Specialization to the Linear Quadratic Regulator (LQR) problem

Tue Nov 19
Ch. 10.3-5

LQR problem (continued)
Riccati differential equation (RDE)
LQR for LTI infinite horizon: Algebraic Riccati equation (ARE)
Properties of LQR inferred from stabilizability, detectability, observability

Thu Nov 21
Ch. 10.4-6

Tools for design and analysis of LQR: Hamiltonian matrix -- eigenvalues and stable subspace
Return difference equation and its use for closed loop LQR analysis
Symmetric root locus; sensitivity of LQR

Tue Dec 3
Ch. 11

Intro to minimum principle
Lagrange multiplier
LQR again

Tue Dec 5
Ch. 11

Bang bang control
nonlinear examples

Tue Dec 10

Review

Tue Dec 17, 7-10 pm in 3017 ECEB
Final eam

The exam will cover material from throughout the semester. There will only be slight emphasis on the optimization material from problem sets 10-12, Chapters 10 and 11, which were topics not included in Exam 2. We did not cover the topic of reduced order observers. To study for the exam I recommend you first review all problem sets and solutions -- comparing course solutions with your own. Then review the reading and your notes. There are no practice exams available. You may bring two sheets of notes two sided to consult during the exam with font size 10 or larger or equivalent handwriting size.