The schedule will be updated and revised as the course progresses.

- 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

- 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

- 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!**- 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