EE263 Autumn 2007-08
Stephen Boyd
Lecture 1
Overview
• 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
= A(t)x(t) + B(t)u(t),
dt
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 are
constant, i.e., don’t depend on t
• when there is no input u (hence, no B or D) system is called
autonomous
• 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 more
than 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 study
linear systems?
• most techniques for nonlinear systems are based on linear methods
• methods for linear systems often work unreasonably well, in practice, for
nonlinear systems
• if you don’t understand linear dynamical systems you certainly can’t
understand 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:
3
2
y
1
0
−1
−2
−3
0
50
100
150
200
250
300
350
t
3
2
y
1
0
−1
−2
−3
0
100
200
300
400
500
600
700
800
900
1000
t
• output waveform is very complicated; looks almost random and
unpredictable
• we’ll see that such a solution can be decomposed into much simpler
(modal) components
Overview
1–14
t
0.2
0
−0.2
0
1
50
100
150
200
250
300
350
50
100
150
200
250
300
350
50
100
150
200
250
300
350
50
100
150
200
250
300
350
50
100
150
200
250
300
350
50
100
150
200
250
300
350
50
100
150
200
250
300
350
50
100
150
200
250
300
350
0
−1
0
0.5
0
−0.5
0
2
0
−2
0
1
0
−1
0
2
0
−2
0
5
0
−5
0
0.2
0
−0.2
0
(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
Overview
−1
ydes =
−0.63
0.36
1–16
let’s apply u = ustatic and just wait for things to settle:
u1
0
−0.2
−0.4
−0.6
−0.8
−1
−200
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
t
1000
1200
1400
1600
1800
1000
1200
1400
1600
1800
1000
1200
1400
1600
1800
1000
1200
1400
1600
1800
u2
0.4
0.3
0.2
0.1
0
−0.1
−200
2
t
y1
1.5
1
0.5
0
−200
t
y2
0
−1
−2
−3
−4
−200
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.
0.2
u1
0
−0.2
−0.4
−0.6
0
10
20
0
10
20
0
10
20
0
10
20
30
40
50
60
30
40
50
60
30
40
50
60
30
40
50
60
t
u2
0.4
0.2
0
−0.2
t
y1
1
0.5
0
−0.5
t
y2
0
−0.5
−1
−1.5
−2
−2.5
t
. . . here y converges exactly in 50 sec
Overview
1–18
in fact by using larger inputs we do still better, e.g.
u1
5
0
−5
−5
0
5
10
15
20
25
0
5
10
15
20
25
0
5
10
15
20
25
0
5
10
15
20
25
t
1
u2
0.5
0
−0.5
−1
−1.5
−5
2
t
y1
1
0
−1
−5
t
0
y2
−0.5
−1
−1.5
−2
−5
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
H(s)
w
A/D
y
• 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
u(t)
1
0
−1
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
s(t)
1.5
1
0.5
0
w(t)
1
0
−1
y(t)
1
0
−1
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) . . .
u(t) (solid) and û(t) (dotted)
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
1
2
3
4
5
6
7
8
9
10
t
RMS error 0.03, well below quantization error (!)
Overview
1–24