ECE 515: Lecture Schedule

System modeling and analysis

Tue Aug 22
Ch. 1

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

Thu Aug 24
Ch. 1

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

Tue Aug 29
Ch. 2

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

Thu Aug 31
Ch. 2

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

Tue Sep 5
Ch. 3

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

Thu Sep 7
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 12
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 14
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 19
Ch. 5

Controllability
Motivation and definition
The general LTV case: the controllability Gramian

Thu Sep 21
Ch. 5

Controllability (cont.)
Controllability of linear time-invariant systems
Controllability matrix, rank criterion

Tue Sep 26
Thu Sep 28

No class: Allerton conference

Tue Oct 3
Ch. 5, 6

Controllability (cont.) and intro to observability
Other tests for controllability
Modal form, the Hautus-Rosenbrock criterion
Observability: motivation and definition
The observability matrix
The general LTV case: the observability Gramian

Thu Oct 5
Ch. 6

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

Feedback

Tue Oct 10
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 12

No class

Tue Oct 17
Ch. 7

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

Thu Oct 19

System invariants
Similarity and feedback equivalence
Canonical forms revisited
Controllability indices

Tue Oct 24
Ch. 8

Tracking and disturbance rejection
Internal model principle
Conditions in terms of controllability
Transfer function approach: Sylvester systems

Tue Oct 26
Ch. 9

Internal model principle revisited
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

Tue Oct 30

In-class review and Q&A

Thu Nov 1

No class: Take-home midterm!

Optimal Control

Tue Nov 7
Ch. 10

Dynamic progamming
Formulation of the finite-horizon optimal control problem
Bellman's dynamic programming principle
The Hamilton-Jacobi-Bellman equation
The Linear Quadratic Regulator (LQR) problem: formulation and derivation of optimal control using completion of squares