EE263 Autumn 2007-08 Stephen Boyd

Lecture 1Overview

• course mechanics

• outline & topics

• what is a linear dynamical system?

• why study linear systems?

• some examples

1–1

Course mechanics

• all class info, lectures, homeworks, announcements on class web page:

www.stanford.edu/class/ee263

course requirements:

• weekly homework

• takehome midterm exam (date TBD)

• takehome final exam (date TBD)

Overview 1–2

Prerequisites

• exposure to linear algebra (e.g., Math 103)

• exposure to Laplace transform, differential equations

not needed, but might increase appreciation:

• control systems

• circuits & systems

• dynamics

Overview 1–3

Major topics & outline

• linear algebra & applications

• autonomous linear dynamical systems

• linear dynamical systems with inputs & outputs

• basic quadratic control & estimation

Overview 1–4

Linear dynamical system

continuous-time linear dynamical system (CT LDS) has the form

dx

dt= A(t)x(t) + B(t)u(t), y(t) = C(t)x(t) + D(t)u(t)

where:

• t ∈ R denotes time

• x(t) ∈ Rn is the state (vector)

• u(t) ∈ Rm is the input or control

• y(t) ∈ Rp is the output

Overview 1–5

• A(t) ∈ Rn×n is the dynamics matrix

• B(t) ∈ Rn×m is the input matrix

• C(t) ∈ Rp×n is the output or sensor matrix

• D(t) ∈ Rp×m is the feedthrough matrix

for lighter appearance, equations are often written

ẋ = Ax + Bu, y = Cx + Du

• CT LDS is a first order vector differential equation

• also called state equations, or ‘m-input, n-state, p-output’ LDS

Overview 1–6

Some LDS terminology

• most linear systems encountered are time-invariant: A, B, C, D areconstant, i.e., don’t depend on t

• when there is no input u (hence, no B or D) system is calledautonomous

• very often there is no feedthrough, i.e., D = 0

• when u(t) and y(t) are scalar, system is called single-input,single-output (SISO); when input & output signal dimensions are morethan one, MIMO

Overview 1–7

Discrete-time linear dynamical system

discrete-time linear dynamical system (DT LDS) has the form

x(t + 1) = A(t)x(t) + B(t)u(t), y(t) = C(t)x(t) + D(t)u(t)

where

• t ∈ Z = {0,±1,±2, . . .}

• (vector) signals x, u, y are sequences

DT LDS is a first order vector recursion

Overview 1–8

Why study linear systems?

applications arise in many areas, e.g.

• automatic control systems

• signal processing

• communications

• economics, finance

• circuit analysis, simulation, design

• mechanical and civil engineering

• aeronautics

• navigation, guidance

Overview 1–9

Usefulness of LDS

• depends on availability of computing power, which is large &increasing exponentially

• used for

– analysis & design– implementation, embedded in real-time systems

• like DSP, was a specialized topic & technology 30 years ago

Overview 1–10

Origins and history

• parts of LDS theory can be traced to 19th century

• builds on classical circuits & systems (1920s on) (transfer functions. . . ) but with more emphasis on linear algebra

• first engineering application: aerospace, 1960s

• transitioned from specialized topic to ubiquitous in 1980s(just like digital signal processing, information theory, . . . )

Overview 1–11

Nonlinear dynamical systems

many dynamical systems are nonlinear (a fascinating topic) so why studylinear systems?

• most techniques for nonlinear systems are based on linear methods

• methods for linear systems often work unreasonably well, in practice, fornonlinear systems

• if you don’t understand linear dynamical systems you certainly can’tunderstand nonlinear dynamical systems

Overview 1–12

Examples (ideas only, no details)

• let’s consider a specific system

ẋ = Ax, y = Cx

with x(t) ∈ R16, y(t) ∈ R (a ‘16-state single-output system’)

• model of a lightly damped mechanical system, but it doesn’t matter

Overview 1–13

typical output:

0 50 100 150 200 250 300 350−3

−2

−1

0

1

2

3

0 100 200 300 400 500 600 700 800 900 1000−3

−2

−1

0

1

2

3

yy

t

t

• output waveform is very complicated; looks almost random andunpredictable

• we’ll see that such a solution can be decomposed into much simpler(modal) components

Overview 1–14

0 50 100 150 200 250 300 350−0.2

0

0.2

0 50 100 150 200 250 300 350−1

0

1

0 50 100 150 200 250 300 350−0.5

0

0.5

0 50 100 150 200 250 300 350−2

0

2

0 50 100 150 200 250 300 350−1

0

1

0 50 100 150 200 250 300 350−2

0

2

0 50 100 150 200 250 300 350−5

0

5

0 50 100 150 200 250 300 350−0.2

0

0.2

t

(idea probably familiar from ‘poles’)

Overview 1–15

Input design

add two inputs, two outputs to system:

ẋ = Ax + Bu, y = Cx, x(0) = 0

where B ∈ R16×2, C ∈ R2×16 (same A as before)

problem: find appropriate u : R+ → R2 so that y(t) → ydes = (1,−2)

simple approach: consider static conditions (u, x, y constant):

ẋ = 0 = Ax + Bustatic, y = ydes = Cx

solve for u to get:

ustatic =(

−CA−1B)

−1ydes =

[

−0.630.36

]

Overview 1–16

let’s apply u = ustatic and just wait for things to settle:

−200 0 200 400 600 800 1000 1200 1400 1600 1800

0

0.5

1

1.5

2

−200 0 200 400 600 800 1000 1200 1400 1600 1800−4

−3

−2

−1

0

−200 0 200 400 600 800 1000 1200 1400 1600 1800−1

−0.8

−0.6

−0.4

−0.2

0

−200 0 200 400 600 800 1000 1200 1400 1600 1800−0.1

0

0.1

0.2

0.3

0.4

u1

u2

y 1y 2

t

t

t

t

. . . takes about 1500 sec for y(t) to converge to ydes

Overview 1–17

using very clever input waveforms (EE263) we can do much better, e.g.

0 10 20 30 40 50 60−0.5

0

0.5

1

0 10 20 30 40 50 60−2.5

−2

−1.5

−1

−0.5

0

0 10 20 30 40 50 60

−0.6

−0.4

−0.2

0

0.2

0 10 20 30 40 50 60−0.2

0

0.2

0.4

u1

u2

y 1y 2

t

t

t

t

. . . here y converges exactly in 50 sec

Overview 1–18

in fact by using larger inputs we do still better, e.g.

−5 0 5 10 15 20 25

−1

0

1

2

−5 0 5 10 15 20 25

−2

−1.5

−1

−0.5

0

−5 0 5 10 15 20 25−5

0

5

−5 0 5 10 15 20 25

−1.5

−1

−0.5

0

0.5

1

u1

u2

y 1y 2

t

t

t

t

. . . here we have (exact) convergence in 20 sec

Overview 1–19

in this course we’ll study

• how to synthesize or design such inputs

• the tradeoff between size of u and convergence time

Overview 1–20

Estimation / filtering

u w yH(s) A/D

• signal u is piecewise constant (period 1 sec)

• filtered by 2nd-order system H(s), step response s(t)

• A/D runs at 10Hz, with 3-bit quantizer

Overview 1–21

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10−1

0

1

0 1 2 3 4 5 6 7 8 9 10−1

0

1

0 1 2 3 4 5 6 7 8 9 10−1

0

1

s(t)

u(t

)w

(t)

y(t

)

t

problem: estimate original signal u, given quantized, filtered signal y

Overview 1–22

simple approach:

• ignore quantization

• design equalizer G(s) for H(s) (i.e., GH ≈ 1)

• approximate u as G(s)y

. . . yields terrible results

Overview 1–23

formulate as estimation problem (EE263) . . .

0 1 2 3 4 5 6 7 8 9 10−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

u(t

)(s

olid

)an

dû(t

)(d

ott

ed)

t

RMS error 0.03, well below quantization error (!)

Overview 1–24