LINEAR SYSTEMS II
ECE230B/ME243B — Winter 07

4 units

Mo We 10:00-11:50pm @ Phelps 1431

1 Course summary

This course complements ECE230A—Linear Systems I in providing the students with the basic tools of modern linear systems theory. The specific courses covered in this course include:

· Multivariable poles and zeros: Smith-McMillan form, poles, transmission and invariant zeros, McMillan degree & minimal realizations

· LQR/LQG control: Algebraic Riccati Equation (ARE), Kalman’s inequality, frequency-domain properties of LQR, loop-shaping using LQR, the cheap control, problem, Minimum Energy (ME) state estimators, Stochastic LQG, Loop transfer recovery (LTR)

· Operator approach to systems theory: frequency-domain transforms, time-invariance, causality, Small-gain Theorem.

· Model reduction: balanced realizations, Hankel operators, balanced truncation

· Feedback stabilization, parameterization of all stabilizing controllers, controller design by convex optimization methods

·Internal model principle: asymptotic tracking and disturbance rejections, necessity and sufficiency of the internal model, tracking of general references.

·Fundamental limits: interlacing (simultaneous stabilization/ strong stabilization)

The students will also be introduced to the computational tools for linear systems theory available in MATLAB.

The intended audience for this course includes, but is not restricted to, students in circuits, communications, control, signal processing, physics, and mechanical and chemical engineering.

A detailed list of the topics covered is available on the web.

2 Instrutor

João P. Hespanha

email: hespanha@ece.ucsb.edu
phone: (805) 893-7042
office: Engineering I, 5157

Office hours: Please email or phone me in advance to schedule for an appointment.
Preferred times are Tue, Thu 2:30-4:00pm

3 Prerequisites

ECE 230A/ME243A Linear Systems I (or consent of instructor)

Basic tools of modern linear systems theory: stability, controllability, observability, realization theory, state feedback, state estimation, separation theorem, etc.

4 Course's Web Page

All information relevant to the course will be continuously posted at the course's web page. The URL is

http://www.ece.ucsb.edu/~hespanha/ece230b/

5 Projects

Students will be assigned two control design projects, which will be the basis for the course grade. Students are expected to turn in a report for each project.

6 Textbook

The course will follow a set of lecture notes that will be made available on web at

http://www.ece.ucsb.edu/~hespanha/published/linearsystems.pdf

Other recommended textbooks are:

[1] G. Dullerud, F. Paganini. A Course in Robust Control Theory. Springer, 1999. (ISBN 0-387-98945-5).

[2] P. Antsaklis, A. Michel. Linear Systems. McGraw Hill, 1997.

[3] J. Maciejowski. Multivariable Feedback Design. Addison-Wesley, 1989. ISBN 0-201-18243-2

[4] H. Kwakernaak, R. Sivan. Linear Optimal Control Systems. John Wiley & Sons, 1972.

7 Study Guide

The following is a tentative schedule for the course. If revisions are needed they will be posted on the course's web page. The rightmost column of the schedule contains the recommended reading for the topics covered on each class. Students are strongly encouraged to read these materials prior to the class.

Class	Content	References
Lect #1 1/8/2007	Introduction and course overview Review of Linear Systems I	syllabus
Lect # 2 1/10/2007	Multi-variable poles and zeros Smith-McMillan form · Smith-McMillan form of a transfer matrix · McMillan degree, poles, and zeros of a transfer matrix · Transmission blocking property · System inverses	Chapter 18 of the lecture notes
1/15/2007	Martin Luther King day
Lect #3 1/17/2007	Minimal realizations · Poles of transfer functions vs. eigenvalues of state-space realizations · Transmission zeros of transfer functions vs. invariant zeros of state-space realizations · Minimal realizations (review from ECE230A/ME243A) · Dimension of minimal realizations for multi-variable transfer matrices	Chapter 19 of the lecture notes or Secs. 3.5, 5.3C of [2]
Lect #4 1/22/2007	LQR/LQG control Linear Quadratic regulation (LQR) · Deterministic Linear Quadratic Regulation (LQR) problem · Solution to the LQR problem · LQR in Matab · Optimal set-point	Chapter 20 of the lecture notes or Secs. 5.1-5.2 of [3], (Secs. 3.3-3.4 of [4])
Lect #5 1/24/2007	Algebraic Riccati Equation (ARE) · Hamiltonian matrix · Domain of the Riccati operator · Stable subspace of the Hamiltonian	Chapter 21 of the lecture notes or Sec. 6.2 of [1]
Let #6 1/29/2007	Frequency-domain and asymptotic properties of LQR · Kalman's inequality · Loop-shaping using LQR · Closed-loop poles for the cheap control asymptotic case · Quadratic cost for the cheap control asymptotic case	Chapter 22 of the lecture notes or Sec. 5.3 of [3] and Sec. 3.8 of [4] Handout with LQG/LQR design example.
Lect #7 1/31/2007	Output-feedback · Deterministic Minimum-Energy (ME) state estimation · Solution to the ME problem · Stochastic Linear Quadratic Gaussian (LQG) state estimation · LQG/LQR output feedback · Loop transfer recovery (LTR)	Chapter 23 of the lecture notes or Sec. 5.4 of [3] (Sec. 4.3 of [4])
Lect #8 1/31/2007	Operator Approach to systems theory Operators · Banach and Hilbert spaces · Operators and Banach algebras	Secs. 3.1-3.2 of [1]
Lect #9 2/5/2007	· Small gain theorem · Spectrum of an operator	Sec. 3.3 of [1]
Lect #10 2/7/2007	Adjoint operator · self-adjoint · positive definite · unitary and isometric operators	Sec. 3.4 of [1]
Lect #11 2/12/2007	No class
Lect #12 2/14/2007	No class
2/19/2007	President’s day
Lect #13 2/21/2007	Transforms · Fourier transform · Laplace Transform	Secs. 4-4.4 of [1]
Lect #14 2/26/2007	Frequency-domain multiplication operators · Multiplication operators · Time-invariance · Causality	Secs. 4.5-4.7 of [1]
Lect #15 2/28/2007	Model reduction Balanced realizations · Lyapunov equations and inequalities · Observability operator and Gramian · Balanced realizations	Secs 5-5.2 of [1]
Lect #16 3/5/2007	Model reduction · Hankel operators · Model reduction · Balanced truncations · Generalized Gramians and truncations	Sec 5.3 of [1]
Lect #17 3/7/2007	Feedback stabilization Stabilizing feedback controllers · General feedback connection · Stability of feedback connections · Stabilizability	Chapter 24 of the lecture notes
Lect #18 3/12/2007	Parameterization of stabilizing controllers · Stable process case ·

Class

Content

References

Lect #1

1/8/2007

Introduction and course overview

Review of Linear Systems I

syllabus

Lect # 2

1/10/2007

Multi-variable poles and zeros

Smith-McMillan form

· Smith-McMillan form of a transfer matrix

· McMillan degree, poles, and zeros of a transfer matrix

· Transmission blocking property

· System inverses

Chapter 18 of the lecture notes

1/15/2007

Martin Luther King day

Lect #3

1/17/2007

Minimal realizations

· Poles of transfer functions vs. eigenvalues of state-space realizations

· Transmission zeros of transfer functions vs. invariant zeros of state-space realizations

· Minimal realizations (review from ECE230A/ME243A)

· Dimension of minimal realizations for multi-variable transfer matrices

Chapter 19 of the lecture notes

Secs. 3.5, 5.3C of [2]

Lect #4

1/22/2007

LQR/LQG control

Linear Quadratic regulation (LQR)

· Deterministic Linear Quadratic Regulation (LQR) problem

· Solution to the LQR problem

· LQR in Matab

· Optimal set-point

Chapter 20 of the lecture notes

Secs. 5.1-5.2 of [3],

(Secs. 3.3-3.4 of [4])

Lect #5

1/24/2007

Algebraic Riccati Equation (ARE)

· Hamiltonian matrix

· Domain of the Riccati operator

· Stable subspace of the Hamiltonian

Chapter 21 of the lecture notes

Sec. 6.2 of [1]

Let #6

1/29/2007

Frequency-domain and asymptotic properties of LQR

· Kalman's inequality

· Loop-shaping using LQR

· Closed-loop poles for the cheap control asymptotic case

· Quadratic cost for the cheap control asymptotic case

Chapter 22 of the lecture notes

Sec. 5.3 of [3] and Sec. 3.8 of [4]

Handout with LQG/LQR design example.

Lect #7

1/31/2007

Output-feedback

· Deterministic Minimum-Energy (ME) state estimation

· Solution to the ME problem

· Stochastic Linear Quadratic Gaussian (LQG) state estimation

· LQG/LQR output feedback

· Loop transfer recovery (LTR)

Chapter 23 of the lecture notes

Sec. 5.4 of [3]

(Sec. 4.3 of [4])

Lect #8

1/31/2007

Operator Approach to systems theory

Operators

· Banach and Hilbert spaces

· Operators and Banach algebras

Secs. 3.1-3.2 of [1]

Lect #9

2/5/2007

· Small gain theorem

· Spectrum of an operator

Sec. 3.3 of [1]

Lect #10

2/7/2007

Adjoint operator

· self-adjoint

· positive definite

· unitary and isometric operators

Sec. 3.4 of [1]

Lect #11

2/12/2007

No class

Lect #12

2/14/2007

No class

2/19/2007

President’s day

Lect #13

2/21/2007

Transforms

· Fourier transform

· Laplace Transform

Secs. 4-4.4 of [1]

Lect #14

2/26/2007

Frequency-domain multiplication operators

· Multiplication operators

· Time-invariance

· Causality

Secs. 4.5-4.7 of [1]

Lect #15

2/28/2007

Model reduction

Balanced realizations

· Lyapunov equations and inequalities

· Observability operator and Gramian

· Balanced realizations

Secs 5-5.2 of [1]

Lect #16

3/5/2007

Model reduction

· Hankel operators

· Model reduction

· Balanced truncations

· Generalized Gramians and truncations

Sec 5.3 of [1]

Lect #17

3/7/2007

Feedback stabilization

Stabilizing feedback controllers

· General feedback connection

· Stability of feedback connections

· Stabilizability

Chapter 24 of the lecture notes

Lect #18

3/12/2007

Parameterization of stabilizing controllers

· Stable process case