Outline
Introduction to Time-Delay Systems
e
Course info
Time-delay systems in control applications
sh
System-theoretic preliminaries
Basic properties
lecture no. 1
Leonid Mirkin
Faculty of Mechanical Engineering, Technion—Israel Institute of Technology
Department of Automatic Control, Lund University
Delay systems in the frequency domain
Rational approximations of time delays
State space of delay systems
Modal properties of delay systems
Stability of transfer functions and roots of characteristic equations
General info
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ECTS credits: 7.5
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Prerequisite: any advanced linear control course
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Grading policy: homework 100% (solutions are graded on a scale of
0–100, each must be at least e 23:1407, average grade of 5 best out
of 6 assignments must be at least 67)
Syllabus
1. Introduction
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Homework solutions are to be submitted electronically to mirkin@control.lth.se
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Literature:
1. My slides.
2. J. E. Marshall, H. Górecki, A. Korytowski, and K. Walton, Time-Delay
Systems: Stability and Performance Criteria with Applications. London:
Ellis Horwood, 1992.
3. K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems.
Boston: Birkhäuser, 2003.
4. R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional
Linear Systems Theory. New York: Springer-Verlag, 1995.
time delays in engineering applications; system-theoretic preliminaries
2. Mathematical modeling of time-delay systems
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frequency domain & modal analyses; state space; rational approximations
3. Stability analysis
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stability notions; frequency sweeping; Lyapunov’s method
4. Stabilization methods
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fixed-structure controllers; finite spectrum assignment; coprime factorization
5. Dead-time compensation
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Smith predictor and its modifications; implementation issues
6. Handling uncertain delays
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Lyapunov-based methods; unstructured uncertainty embedding
7. Optimal control and estimation (??)
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H 2 optimizations
On a less formal side
Outline
What this course is about. . .
Course info
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system-theoretic and control aspects of delays in dynamical systems
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exploiting the structure of the delay element
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giving a flavor of ideas in the field
System-theoretic preliminaries
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showing that things are (relatively) simple if the viewpoint is right
Basic properties
Time-delay systems in control applications
Delay systems in the frequency domain
. . . and what it isn’t
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digging up most general and mathematically intriguing cases
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answering the very problem that motivated you to take this course
Rational approximations of time delays
State space of delay systems
Modal properties of delay systems
Stability of transfer functions and roots of characteristic equations
Time delay element
Why delays?
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Ubiquitous in physical processes
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y D Dh u W
y.t/ D u.t
h/
t Ch t !
h
y
Dh
t
0
I
u
or
yN D DN h uN W
yŒk D uŒk
kCh k !
h
h
yN
DN h
I
t!
0
k
uN
loop delays
process delays
...
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Compact/economical approximations of complex dynamics
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Exploiting delays to improve performance
k!
Loop delays: steel rolling
Loop delays: networked control
Plant
Network
Controller
Sampling, encoding, transmission, decoding need time. This gives rise to
Thickness can only be measured at some distance from rolls, leading to
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measurement delays
Loop delays: temperature control
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measurement delays
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actuation delays
Internal delays: regenerative chatter in metal cutting
Waves from
previous pass
y(t-T)
Workpiece
Waves from
current vibration
y(t)
X
Ft
+
Chip
h0
Y
Tool
h0
a
Fc
Chip
Y
X
y(t-T)
h0+dy
Tool
Feed
Direction
Ft
h0
y(t)
Tool
Fc
Deviation from the ideal cut satisfies
my.t
R / C c y.t
P / C ky.t / D Kf a h0 C y.t
Everybody experienced this, I guess. . .
where T is time of one full rotation of workpiece.
T/
y.t / ;
Delays as modeling tool: heating a can
Delays as modeling tool: torsion of a rod
Transfer function of a heated can (derived from PDE model):
Transfer function of a free-free uniform rod at distance x from actuator:
G.s/ D
J0
q1
s
R
˛
1
X
G.s/ D
s
2
q 1
s
L :
mR J1 .mR/ sC˛2m
cosh
C2m ˛
2
mD1
C
Its approximation by G2 .s/ D
1
e sh
.1 sC1/.2 sC1/
k e
s
Its approximation by Gr .s/e
is reasonably accurate:
xhs
Ce
1 e
xhs
.2 x/hs
;
2hs
does capture high-frequency phase lag:
Bode Diagram
0
0.7
−0.1
0.6
−0.2
−0.3
0.4
−0.4
0.3
−0.5
0.2
−0.6
0.1
−0.7
30
20
20
10
0
−10
−20
1000
2000
3000
4000
5000
6000
time (s)
−0.8
−0.4
0
0.2
0.4
0.6
0.8
1
−20
−30
−40
−50
0
−50
0
−720
−1440
−2160
−3600
0
10
−0.2
0
−10
−40
−2880
0
0
10
−30
−720
Phase (deg)
0.5
40
30
Magnitude (dB)
0.1
0.8
Bode Diagram
40
Phase (deg)
0.9
Magnitude (dB)
0.2
imag
y(t) (°C)
1
0 x 1:
−1440
−2160
−2880
−3600
0
10
1
10
Frequency (rad/sec)
1
10
Frequency (rad/sec)
real
Exploiting delays: repetitive control
Exploiting delays: preview control
Any T -periodic signal f .t/ satisfies (with suitable initial conditions)
If we know reference in advance, we can exploit it to improve performance:
f .t/
f .t
T/ D 0
or, in frequency domain,
f .s/ D
1
1
e
min
0:707
sT
This motivates the configuration, called repetitive control:
sensor
y
P .s/
u
e
C.s/
e
r
11111
00000
00000
11111
m
sT
available preview
which is
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generalization of the internal model controllers (including I)
and guarantees (if stable!)
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asymptotically perfect tracking of any T -periodic reference r
0
preview length
Outline
Linear systems
We think of systems as linear operators between input and output signals:
Course info
y D G u:
Time-delay systems in control applications
G is said to be
System-theoretic preliminaries
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Basic properties
causal if ˘ G .I
˘ / D 0, 8 , where ˘ is truncation at time t!
y
Delay systems in the frequency domain
Rational approximations of time delays
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State space of delay systems
t Ch t !
y
Modal properties of delay systems
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u
time-invariant if it commutes with shift operator: GSh D Sh G , 8h
h
Stability of transfer functions and roots of characteristic equations
t!
˘
Sh
t!
t
0
u
stable if 9 0 (independent of u) such that kG uk kuk, 8u
LTI systems
Transfer functions
Let g.t/ be impulse response of an LTI system G . Then
Z 1
y.t/ D
g.t /u. /d
(convolution integral)
In the s -domain (Laplace transform domain) convolution becomes product:
For causal systems g.t/ D 0 whenever t < 0. Then
where G.s/ D Lfg.t /g ´
y.t / D
1
y.t/ D
or even
y.t/ D
t
Z
g.t
/u./d
1
Z
t
g.t
/u. /d
/u./d ” y.s/ D G.s/u.s/;
g.t
0
R1
0
g.t /e
st
dt is called the transfer function of G .
Some basic definitions:
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Static gain: G.0/
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Frequency response: G. j!/ ´ G.s/jsD j!
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0
if we assume that u.t/ D 0 whenever t < 0.
t
Z
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High-frequency gain: lim sup!!1 kG. j!/k
G.s/ is said to be proper if 9˛ > 0 such that sups2C˛ G.s/ < 1, where
C˛ ´ fs 2 C W Re s > ˛g D
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˛
G.s/ is said to be strictly proper if lim˛!1 sups2C˛ G.s/ D 0
Transfer functions: causality & L2 -stability
Rational transfer functions
Theorem
A p q transfer function G.s/ is said to be rational if 8i D 1; p and j D 1; q
LTI G is causal iff its transfer function G.s/ is proper.
Gij .s/ D
Some definitions:
R1
1=2
(kf
is energy of f .t /)
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L -norm of f .t/: kf k2 ´
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G is said to be L2 stable if kG kL2 7!L2 ´ supkuk2 D1 kG uk2 < 1
˚
H 1 ´ G.s/ W G.s/ analytic in C0 and sups2C0 kG.s/k < 1
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2
0
2
kf .t /k dt
k22
Theorem
LTI G is causal and L2 -stable iff G 2 H 1 . Moreover, in this case
kG kL2 7!L2 D kGk1 ´ sups2C0 kG.s/k D sup!2R kG. j!/k:
b mij s mij C C b1 s C b0
anij s nij C C a1 s C a0
for some nij ; mij 2 ZC .
System G is finite dimensional iff its transfer function G.s/ is rational.
Some properties greatly simplified when G.s/ is rational:
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rational G.s/ is proper (strictly proper) iff nij mij (nij > mij ) 8i; j
rational G 2 H 1 iff G.s/ proper and has no poles in C0 ´ jR [ C0
high-frequency gain of rational G.s/ is kG.1/k
Note that G 2 H 1 implies that G.s/ is proper.
State-space realizations
State-space realizations (contd)
Any causal LTI finite-dimensional system G admits state space realization in
the sense that there are n 2 ZC and x.t / 2 Rn (called state vector) such that
(
x.t/
P
D Ax.t / C Bu.t /; x.0/ D x0
y D Gu )
y.t/ D C x.t/ C Du.t /
Assume that x0 D 0 (no history to accumulate). Then:
for some A 2 R
x0 . In this case
nn
,B2R
nm
,C 2R
x.t C / D eA x.t / C
pn
,D2R
pm
, and initial condition
Z
eA.
0
/
Bu.t C /d ;
implying that
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if we know x.t/ and future inputs, we can calculate future outputs
(i.e., no knowledge of past inputs required). This means that
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state vector is history accumulator,
which is the defining property of state vector.
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impulse response of G is g.t / D Dı.t / C C eAt B ,
transfer function of G is G.s/ D D C C.sI
A B
A/ B µ
C D
1
State-space realization is not unique. There even are realizations of the very
same system with different state dimensions. A realization called
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minimal if no other realizations of lower dimension exist
and .A; B; C; D/ is minimal iff .A; B/ controllable and .C; A/ observable.
Any two minimal realizations connected via similarity transformation:
x.t / 7! T x.t /
)
A B
TAT
7
!
C D
CT
1
1
TB
;
D
which changes neither impulse response nor transfer function, obviously.
System properties via state-space realizations
State-space calculus
An LTI causal finite-dimensional system is
These can be verified via simple flow-tracing:
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L2 stable iff its minimal realization has Hurwitz “A” matrix1 .
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Addition:
This is independent of the realization chosen.
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The “D ” matrix is informative too:
A B
I G.s/ D
is strictly proper iff D D 0
C D
A B
I High-frequency gain of G.s/ D
is exactly kDk
C D
A2 B2
A1 B1
C
C2 D1
C2 D2
A2 B2
C2 D2
A1 B1
C1 D1
2
3
A1 0
B1
5
B2
D 4 0 A2
C1 C2 D1 C D2
2
3 2
3
A1
0
B1
A2 B2 C1 B2 D1
A1
B1 5
D 4 B2 C1 A2 B2 D1 5 D 4 0
D2 C1 C2 D2 D1
C2 D2 C1 D2 D1
Inverse (exists iff det D ¤ 0):
1A
Multiplication:
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A B
C D
1
D
A
BD 1 C BD 1
D 1C
D 1
matrix is said to be Hurwitz if it has no eigenvalues in C0 (closed right-half plane).
Schur complement
Outline
If det M11 ¤ 0,
M11 M12
I
0
M11 0
I M111 M12
M D
D
;
M21 M22
M21 M111 I
0 11
0
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while if det M22 ¤ 0,
M11 M12
I M12 M221
22 0
I
0
M D
D
;
0
I
M21 M22
0 M22
M221 M21 I
where
Course info
Time-delay systems in control applications
System-theoretic preliminaries
Basic properties
Delay systems in the frequency domain
Rational approximations of time delays
11 ´ M22
M21 M111 M12
and
22 ´ M11
M12 M221 M21
are Schur complements of M11 and M22 , respectively. Consequently,
det M D det M11 det 11 D det M22 det 22 ;
provided corresponding invertibility holds true.
State space of delay systems
Modal properties of delay systems
Stability of transfer functions and roots of characteristic equations
Time delay element: continuous time
Transfer function of Dh
By the time shifting property of the Laplace transform:
y D Dh u W
t Ch t !
h
y.t/ D u.t
h/
y
Dh
t
0
t!
y.t / D u.t
u
h/ ” y.s/ D e
sh
u.s/
Thus
Causality follows by
Dh .s/ D e
Dh ˘ D ˘ Ch Dh
)
˘ / D ˘ .I
˘ Dh .I
˘ / D .˘
Transfer function of Dh is
˘ /Dh D 0
Time-invariance follows by
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entire (i.e., analytic in whole C)
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bounded in C˛ for every ˛ 2 R
Hence,
Dh S D DhC D S Dh
Impulse response is clearly ı.t
;
which is irrational.
˘Ch /Dh :
Since ˘ ˘Ch D ˘ , we have that
˘ Dh .I
sh
e
sh
2 H1
h/
Time delay element: discrete time
Outline
Course info
yN D DN h uN W
yŒk
N D uŒk
N
kCh k !
h
h
yN
DN h
0
k
k!
uN
Time-delay systems in control applications
System-theoretic preliminaries
By the time shifting property of the ´-transform:
yŒk
N D uŒk
N
Thus
h ” y.´/
N
D´
Basic properties
h
u.´/
N
1
DN h .´/ D h ;
´
which is rational (of order h, all h poles are at the origin, no finite zeros).
Delay systems in the frequency domain
Rational approximations of time delays
State space of delay systems
Modal properties of delay systems
Stability of transfer functions and roots of characteristic equations
Frequency response
Effects of I/O delay on rational transfer functions
Obviously,
Consider the simplest interconnection:
e
Has
sh
jsD j! D e
j!h
D cos.!h/
j sin.!h/
L.s/ D Lr .s/e
unit magnitude (je j!h j 1) and
linearly decaying phase (arg e j!h D !h, in radians if ! —in rad/sec)
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Bode Diagram
0
0
−180
for some rational Lr .s/:
j!h ,
and
meaning that
arg L. j!/ D arg Lr . j!/
In other words, delay in this case
0
Open−Loop Gain (dB)
Magnitude
jL. j!/j D jLr . j!/j
Nichols Chart
2
Phase (deg)
In this case L. j!/ D Lr . j!/e
sh
−2
I
does not change the magnitude of Lr . j!/ and
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adds phase lag proportional to ! .
−4
−6
−360
−8
−540
0
1
10
−540
10
Frequency (rad/sec)
−360
−180
Open−Loop Phase (deg)
0
Effects of I/O delay: Nyquist diagram
0
0
Imaginary Axis
Magnitude (dB)
Effects of I/O delay: Bode diagram
Phase (deg)
0
−1
10
0
10
Frequency (rad/sec)
1
10
−1
0
Real Axis
!h:
Effects of I/O delay: Nichols chart
It’s not always so easy
If delay not I/O, frequency response plots might be much more complicated.
Consider for example the (stable) system:
0
1
e 2s
;
2s
with impulse response
g.t / D
Bode Diagram
1.t /
1.t
2
2/
;
Nyquist Diagram
0
0
Magnitude
−20
−0.2
−40
Imaginary Axis
Open−Loop Gain (dB)
G.s/ D
Phase (deg)
0
−540
−360
−180
0
Open−Loop Phase (deg)
Outline
Course info
Time-delay systems in control applications
System-theoretic preliminaries
Basic properties
Delay systems in the frequency domain
Rational approximations of time delays
State space of delay systems
Modal properties of delay systems
Stability of transfer functions and roots of characteristic equations
−0.4
−90
−0.6
−180
−1
10
0
10
Frequency (rad/sec)
1
10
−0.2
0
0.2
0.4
Real Axis
0.6
0.8
1
Why to approximate
Delay element is infinite dimensional, which complicates its treatment. It is
not a surprise then that we want to approximate delay by finite-dimensional
(rational) elements to
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use standard methods in analysis and design,
I
use standard software for simulations,
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avoid learning new methods,
I
...
What to approximate: bad news
What to approximate: good news
On the one hand,
Yet we never work over infinite bandwidth. Hence, we
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phase lag of the delay element is not bounded (and continuous in ! ).
On the other hand,
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I
need to approximate e
sh
in finite frequency range
or, equivalently,
rational systems can only provide finite phase lag.
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Therefore, phase error between e
and any rational transfer function R.s/
is arbitrarily large. Moreover, for every R.s/ there always is !0 such that
approximate F .s/e
sh
for low-pass (strictly proper) F .s/.
sh
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arg e
j!h
arg R. j!/ continuously decreasing function of ! , 8! !0 .
Hence there always is frequency !1 such that
arg e
j!1 h
arg R. j!1 / D This, together with the fact that je
I
j!h j
2k
i.e.,
R. j!1 /
1, means that
rational approximation of pure delay, e
sh
I
e
j!1 h
phase lag of delay over finite bandwidth is finite
and
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magnitude of F . j!/e
j!h
decreases as ! increases,
which implies that at frequencies where the phase lag of F . j!/e
the function effectively vanishes.
large,
Also, we may consider h D 1 w.l.o.g., otherwise s ! s= h makes the trick.
we never can get better approximation than with R.s/ D 0 . . .
Truncation-based methods
Truncation-based methods: naı̈ve approach
General idea is to
Note that
I
j!h
, is pretty senseless
as there always will be frequencies at which error2 is 1 (i.e., 100%).
2 Thus,
This can be done, since
truncate some power series,
which could give accurate results in a (sufficiently large) neighborhood of 0.
e s=2
es=2
and truncate Taylor series of numerator and denominator. We could get:
Pn
1
i
i D0 iŠ2i . s/
s
e Pn
1
i
iD0 iŠ2i s
e
s
D
This yields:
n
e
sh
1
2
sh
2
1C sh
2
sh
s 2 h2
2
8
2 2
sh
1C 2 C s 8h
1
1
3
sh
s 2 h2
2 C 8
2 2
sh
1C 2 C s 8h
1
For n D 2 called Kautz formula. But
I
becomes unstable for n > 4 !
4
s 3 h3
48
3 h3
C s 48
sh
s 2 h2
2 C 8
2 2
sh
1C 2 C s 8h
1
s 3 h3
s 4 h4
48 C 384
4 h4
s 3 h3
C 48 C s384
Truncation-based methods: Padé approximation
Example: Œ2; 2-Padé approximation
Consider approximation
In this case RŒ2;2 .s/ D
e
s
Pm .s/
µ RŒm;n .s/;
Qn .s/
s
s2 s3
C
C 1Š
2Š 3Š
0
00
000
RŒm;n
.0/s
RŒm;n
.0/s 2
RŒm;n
.0/s 3
RŒm;n .s/ D RŒm;n .0/ C
C
C
C 1Š
2Š
3Š
s
RŒ2;2 .s/ D 1
D1
and Taylor expansions are
s2
2
2.q13
s3
s4
C
6
24
q1 q0 / 3 2.q14
s C
3
sC
2q 2
2q1
s C 21 s 2
q0
q0
q0
from which
D1
The idea of Œm; n-Padé approximation is to find coefficients of RŒm;n .s/ via
I
s
e
where Pm .s/ and Qn .s/ are polynomials of degrees m and n, respectively.
Taylor expansions at s D 0 of each side are
e
s 2 q1 sCq0
s 2 Cq1 sCq0
q04
s4
q1 2
1
D
4q1
6
and
q0 D 2q1
2q12 q0 /
and then q1 D 6 and q0 D 12, matching 5 coefficients.
Thus, Œ2; 2-Padé approximation is
2
matching first n C m C 1 Taylor coefficients
e
of these two series. If n D m, it can be shown that Pn .s/ D Qn . s/.
s
s
1 2s C 12
s 2 6s C 12
2
D
:
s2
s C 6s C 12
1 C 2s C 12
Truncation-based methods: Padé approximation (contd)
Padé approximation: example
General formula for Œn; n-Padé approximation is
1
Consider Padé approximation of sC1
e s . This can be calculated by Matlab
function pade(tf(1,[1 1],’InputDelay’,1),N).
Pn
e
s
n .2n i/Š
.
i D0 i
.2n/Š
Pn
n .2n i/Š
iD0 i
.2n/Š
s/i
si
Pn
.2n i/ŠnŠ
iD0 .2n/Š.n i /Š i Š .
Pn
.2n i/ŠnŠ
iD0 .2n/Š.n i/Š iŠ
D
s/i
si
s
e
sC1
and its 2nd and 4th order approximations
Corresponding approximation errors
Bode Diagram
This yields:
Bode Diagram
0
0
sh
1
sh
2
1C sh
2
2
s 2 h2
1 sh
2
12
s 2 h2
1C sh
2 C 12
3
s 2 h2
1 sh
2 C 10
s 2 h2
1C sh
2 C 10
4
s 3 h3
120
3 h3
C s120
Using Routh-Hurtitz test one can prove that
I
Œn; n-Padé approximation stable for all n.
s 2 h2
1 sh
2 C 28
s 2 h2
1C sh
2 C 28
s 3 h3
s 4 h4
84 C 1680
3
3
4 h4
h
C s 84
C s1680
−20
−15
−40
−20
Magnitude (dB)
e
1
−25
−30
0
−60
−180
Phase (deg)
n
Magnitude (dB)
−5
−10
−80
−360
−540
−720
−100
−900
−1080
−1260
−1
10
0
10
Frequency (rad/sec)
1
10
−120
−1
10
0
10
Frequency (rad/sec)
1
10
Padé approximation: example (contd)
Padé approximation: example (contd)
We may also compare step responses and Nichols charts
Increasing approximation order improves the match between step responses
1
e s and its Padé approximation:
of sC1
s
e
sC1
and its 2nd and 4th order approximations
Step Response
Nichols Chart
1
e
10
s
sC1
and its 50th order approximation
e
s
and its 50th order approximation
Step Response
Step Response
5
0.8
1.2
1
0
0.8
−10
0.6
0.6
−15
0.2
Amplitude
0.4
1
0.8
−5
Amplitude
Open−Loop Gain (dB)
Amplitude
0.6
0.4
0.4
−20
0.2
0
0.2
−25
0
−0.2
0
0.5
1
1.5
2
Time (sec)
2.5
3
3.5
4
−30
−720
−630
−540
−450
−360
−270
Open−Loop Phase (deg)
−180
From loop shaping perspectives,
I
−90
0
0
−0.2
−0.2
0
0.5
1
1.5
approximation performance depends on crossover requirements.
Outline
Course info
Time-delay systems in control applications
System-theoretic preliminaries
Basic properties
Rational approximations of time delays
2.5
−0.4
4
if input delayed by h
x.t
P / D Ax.t / C Bu.t /
Solution then becomes:
Z
A
x.t C / D e x.t / C
t C
!
eA.tC
/
0
0.5
1
1.5
2
Time (sec)
2.5
3
De
A
De
x.t / C
h/d
!
Bu.
h/d
x.t / C
De
x.t / C
tC
Z
eA.t
/
x.t
P / D Ax.t / C Bu.t
Bu.
t
A
Stability of transfer functions and roots of characteristic equations
3.5
3.5
4
s
!
State equation with input delay
State space of delay systems
Modal properties of delay systems
3
Not true for the approximation of the pure delay e
A
Delay systems in the frequency domain
2
Time (sec)
h/
t
Z
!
t hC
A.t h /
e
Bu. /d
t h
Z
t
e
t h
A.t h /
Bu. /d C
!
t hC
Z
A.t h /
e
Bu. /d
t
It depends on initial “state” x.t /, future inputs over Œt; t h C  (if > h),
and past inputs over Œt h; t  (or Œt h; t h C  if < h).
“In my country there is problem. . . ” (B. Sagdiyev)
Checking intuition on discrete-time case
. . . and that problem is:
Consider
I
xŒk
N C 1 D ANxŒk
N C BN uŒk
N
x.t / no longer accumulates the history.
This, in turn, implies that x.t/ can no longer be regarded as the “state”.
This system can be thought of as serial interconnection
xN
Intuitively, the “true” state vector at every time instance t should contain
I
x.t/
I
u.t C / for all 2 Œ h; 0—denoted u .t /
´ h uN
PN0 .´/
xN
PN0 .´/
´ h uN
where
AN BN
PN0 .´/ D
I 0
DN h .´/
uN
AN
0
60
6
07
7 6
0
:: 7 6
6:
:7
7D6
6 ::
07
7 6
60
I5 6
40
0
I
BN
0
0
::
:
2
0
60
6 :
N BN 6
A
6 ::
PNh .´/ D
6
I 0 60
6
40
I
0 I :: : :
:
:
0 0 0 0 0 0 I
0
::
:
0
0
::
:
I
0
0
3
and
0 I :: : :
:
:
0 0 0 0 I 0 0 0
60
6
6 ::
6
DN h .´/ D 6 :
60
6
40
I
0
::
:
0
0
::
:
3
0
07
7
:: 7
:7
7
I 07
7
0 I5
0 0
Checking intuition on discrete-time case (contd)
To recover the state vector, write state equation:
3 2 N
xŒk
N C 1
A
6 uŒk
7 60
N
h
C
1
6
7 6
6 uŒk
6
h C 2 7
6N
7 60
D
6
7 6 ::
::
6
7 6:
:
6
7 6
4 uŒk
N
1 5 4 0
2
Thus, for PNh .´/ ´ PN0 .´/DN h .´/ we have:
2
uN
DN h .´/
2
Checking intuition on discrete-time case (contd)
h:
3
0
07
7
7
07
:: 7
7
:7
7
0 0 0 I 07
7
0 0 0 0 I 5
0 0 0 0 0
0
I
0
::
:
0
0
I
::
:
::
:
0
0
0
::
:
„
uŒk
N
ƒ‚
xN a ŒkC1
…
BN
0
0
::
:
32
3 2 3
0
xŒk
N
0
6 uŒk
7 607
07
N
h
76
7 6 7
6N
6 7
07
h C 1 7
7 6 uŒk
7 607
C
N
7 6 :: 7 uŒk
6
::
:: 7
6
7 6:7
:7
:
76
7 6 7
0 0 0 I 5 4 uŒk
N
2 5 4 0 5
0 0 0 0 0
uŒk
N
1
I
„
ƒ‚
…
0
I
0
::
:
0
0
I
::
:
::
:
Thus, the state vector at time k , xN a Œk, indeed
I
xN a Œk
includes both xŒk
N
and whole input history uŒi
N in k
hi k
1
State equation with input delay (contd)
State equation with output delay
Thus, the state vector of
Consider
x.t
P / D Ax.t / C Bu.t
x.t
P / D Ax.t / C Bu.t /
h/
at time t is .x.t/; u .t// 2 .R ; fŒ h; 0 7! R g/. This implies (among many
other things) that
n
I
m
´ h xN
initial conditions for this system are .x.0/; u .0//,
which is also a function. For example,
I
and assume that that we measure delayed x , i.e., y.t / D x.t
Discrete counterpart looks like this:
DN h .´/
xN
2
and u. / D 0;
0
60
6
6 ::
6
PNh .´/ D 6 :
60
6
40
I
2
8Œ h; 0:
There is more consistent (and elegant) way to reflect all this via state-space
description, using semigroup formalism. See (Curtain and Zwart, 1995) for
details.
State equation with output delay (contd)
3 2
xŒk
N
h C 1
0
6 xŒk
6
7
h C 2 7 6 0
6N
6
7 6 ::
::
6
7 6:
:
6
7D6
6 xŒk
6
1 7
6 N
7 60
4
5 40
xŒk
N
xŒk
N C 1
0
ƒ‚
…
„
2
0 0
I 0
:: : : ::
: :
:
0 0 I
0 0 0
0 0 0
I
0
::
:
xN a ŒkC1
0 0
I 0
:: : : ::
: :
:
0 0 I
0 0 0
0 0 0
I
0
::
:
0
0
60
6
07
6:
7
6 ::
:: 7 AN BN
6
:7
D6
7
60
I 0
07
6
7
60
5
6
I
40
0
I
3
I
includes whole history of xŒi
N  in k
32
3 2 3
0
0
xŒk
N
h
6
7
6
7
0 7 6 xŒk
N
h C 1 7 6 0 7
7
7 6 :: 7
:: 7 6
::
7
6
6
7
7
: 76
:
N
7 C 6 : 7 uŒk
6
7
6
7
0 7 6 xŒk
N
2 7 6 0 7
7
I 5 4 xŒk
N
1 5 4 0 5
BN
AN
xŒk
N
ƒ‚
…
„
xN a Œk
0
0
0
0
0 0
I 0
:: : :
:
: ::
:
0 I
0 0
0 0
0 0 0
0
::
:
0
I
AN
0
3
0
07
7
:: 7
:7
7
07
7
7
07
7
BN 5
0
Returning to continuous-time case, state vector of
(
x.t
P / D Ax.t / C Bu.t /
y.t / D x.t
h/
at time t is x .t / 2 fŒ h; 0 7! Rn g (may be convenient to write .x.t /; x .t //).
The
I
initial condition is then the function x .0/
x./ D 0;
hi k
I
0
::
:
State equation with output delay (contd)
and zero initial conditions would mean
Thus, the state vector at time k , xN a Œk,
uN
PN0 .´/
with
zero initial conditions should read
x.0/ D 0
h/.
1
8 2 Œ h; 0:
State delay equations
Homogeneous LTI state equations: classification
If we use a P controller u.t/ D Ky.t /, the closed loop system becomes
(Lumped-delay) retarded equation:
x.t
P / D Ax.t / C BKx.t
h/:
x.t
P /D
This kind of equations called retarded functional differential equation.
r
X
Ai x.t
0 D h0 < h1 < < hr D h
hi /;
iD0
(Lumped-delay) neutral equation:
r
X
If we use a D controller u.t/ D K y.t
P /, the closed loop system becomes
x.t/
P
D Ax.t/ C BK x.t
P
or x.t
P /
h/
BK x.t
P
h/ D Ax.t /
iD0
Ei x.t
P
hi / D
r
X
Ai x.t
hi /;
i D0
0 D h0 < h1 < < hr D h
Distributed-delay retarded equation:
This kind of equations called neutral functional differential equation.
x.t
P /D
If ˛./ D
I
P
i
Z
0
h
˛./x.t C /d
Ai ı.t C hi /, we have the lumped-delay equation above.
In all cases the “true” state is x .t / 2 fŒ h; 0 7! Rn g.
Adding inputs and outputs
The same in s domain (with zero initial conditions)
(Still not the most) general form:
(Still not the most) general form:
„
x.t/
P C
Z
0
hx
./x.t
P C /d D
y.t / D
Z
0
Z
hx
0
hx
˛. /x.t C /d C
. /x.t C /d C
Z
0
Z
hu
0
hu
„
ˇ. /u.t C /d
Z
s IC
0
s
hx
./e d X.s/ D
ı. /u.t C /d
Y.s/ D
Z
0
Z
hx
0
s
˛./e dX.s/ C
s
hx
./e dX.s/ C
Z
0
Z
hu
0
ˇ. /es d U.s/
ı. /es dU.s/
hu
with the state “vector” .x .t /; u .t // 2 fŒ hx ; 0 7! Rn g; fŒ hu ; 0 7! Rm g .
with zero initial conditions x./ D 0 ( 2 Œ hx ; 0), u./ D 0 ( 2 Œ hu ; 0).
Important special (lumped-delay) case:
Important special (lumped-delay) case:
†X
rx
iD0
Ei x.t
P
hi / D
y.t / D
rx
X
iD0
rx
X
iD0
Ai x.t
Ci x.t
hi / C
hi / C
ru
X
iD0
ru
X
†
Bi u.t
hi /
s
rx
X
iD0
Di u.t
i D0
with E0 D I and 0 D h0 < h1 < < hmaxfrx ;ru g D h.
hi /
Ei e
shi
X.s/ D
Y .s/ D
rx
X
iD0
rx
X
iD0
Ai e
Ci e
shi
shi
X.s/ C
X.s/ C
ru
X
i D0
ru
X
iD0
with E0 D I and 0 D h0 < h1 < < hmaxfrx ;ru g D h.
Bi e
shi
U.s/
Di e
shi
U.s/
Outline
Characteristic equation
Distributed-delay equation
Z 0
Z
s
s IC
./e d X.s/ D
Course info
Time-delay systems in control applications
hx
System-theoretic preliminaries
0
hx
˛. /es dX.s/ C
Z
0
ˇ. /es dU.s/
hu
(or its lumped-delay counterpart) can be rewritten as
Basic properties
X.s/ D 1 .s/
Delay systems in the frequency domain
Z
0
ˇ./es dU.s/
or X.s/ D 1 .s/
hu
ru
X
Bi e
shi
U.s/;
iD0
where
Rational approximations of time delays
.s/ ´
State space of delay systems
Z
0
hx
s.I C . //
˛./ es d
or .s/ ´
rx
X
.sEi
Ai / e
shi
i D0
called the characteristic matrix. Then equation
Modal properties of delay systems
detŒ.s/ µ .s/ D 0
Stability of transfer functions and roots of characteristic equations
called characteristic equation.
Example 1
Example 2
Consider
0 1
x.t
P /D
x.t/
0 1
0 0
x.t
k1 k2
Consider
0 0
x.t
P /C
x.t
P
k1 k2
h/
Then
0 1
h/ D
x.t /
0 1
Then
1 0
.s/ D s
0 1
0 1
0 0
C
e
0 1
k1 k2
sh
s
D
k1 e
and
sh
1
s C 1 C k2 e
sh
1 0
0 0
.s/ D s
Cs
e
0 1
k1 k2
sh
0 1
s
D
0 1
sk1 e
and
2
.s/ D s C s C .k2 s C k1 /e
sh
.s/ D s 2 C s C .k2 s 2 C k1 s/e
sh
sh
1
s C 1 C sk2 e
sh
Example 3
Quasi-polynomials
Consider
0 1
x.t
P /D
x.t/
0 1
k1 0
x.t
0 k2
General form:
h/
det
Then
rx
X
.sEi
Ai /e
s hQ i
!
iD0
1 0
.s/ D s
0 1
0 1
k1 0
C
e
0 1
0 k2
sh
s C k1 e
D
0
sh
1
s C 1 C k2 e
and
2
.s/ D s C s C .k1 C k2 /s C k1 e
sh
C k1 k2 e
2sh
sh
X
D P .s/ C
Qi .s/e
shi
i
for polynomials P .s/ 6 0 and Qi .s/ (9j , Qj .s/ 6 0) and delays hi > 0.
Classification by delay pattern:
1. single-delay: P .s/ C Q.s/e
sh
2. commensurate-delay: P .s/ C
P
i
Qi .s/e
3. incommensurate-delay: if at least one of
sih
hi
hj
is irrational
Classification by principal degrees of s :
1. retarded: deg P .s/ > deg Qi .s/, 8i
2. neutral: deg P .s/ deg Qi .s/, 8i , and 9j s.t. deg P .s/ D deg Qj .s/
3. advanced: 9j s.t. deg P .s/ < deg Qj .s/
Roots of quasi-polynomials
Roots of quasi-polynomials: where are they located
Apparently, the simplest example is
Some fundamental properties:
1 C ke
sh
;
k > 0:
It is readily seen that it has
I
infinite number of roots,
those at
.1 C 2i /
ln k
Cj
; i 2Z
h
h
N 0 iff k 1 and in C n C
N 0 iff 0 < k < 1).
(all these roots are in C
sD
1. there is a finite number of roots within any finite region of C
(meaning there are no accumulation points for roots of quasi-polynomials)
2. roots for large values of jsj belong to a finite number of areas
ˇ
ˇ
ˇRe s C ˇi lnjsjˇ < for some > 0 and ˇi 2 R. For
I
I
I
retarded quasi-polynomials ˇi > 0
neutral quasi-polynomials ˇi D 0
advanced quasi-polynomials ˇi < 0
3. retarded quasi-polynomials have a finite number of roots in C˛ for all ˛
This is generic property, i.e.,
I
quasi-polynomials have infinite number of roots.
Rightmost root: retarded case
Rightmost root: neutral case
Consider
Consider
x.t/
P
D
rx
X
Ai x.t
hi /;
iD0
x .0/ D . /
rx
X
iD0
and let .s/ be its characteristic quasi-polynomial. Define
r ´ maxfRe s W .s/ D 0g
rx
X
Ai x.t
hi /;
iD0
x .0/ D . /
r ´ supfRe s W .s/ D 0g
Theorem
For any > r there is a > 0 such that
2Œ h;0
hi / D
and let .s/ be its characteristic quasi-polynomial. Define
Theorem
kx.t /k et max k. /k;
Ei x.t
P
8t 2 RC
For any > r there is a > 0 such that
P
kx.t /k et max .k. /k C k./k/;
2Œ h;0
for all continuous initial conditions .
8t 2 RC
for all continuous and differential initial conditions .
Outline
Course info
Simple special case
Special (SISO single-delay) case:
Time-delay systems in control applications
System-theoretic preliminaries
Basic properties
Delay systems in the frequency domain
Rational approximations of time delays
State space of delay systems
Modal properties of delay systems
Stability of transfer functions and roots of characteristic equations
G.s/ D
N0 .s/
M0 .s/ C Mh .s/e
sh
;
for some real polynomials M0 .s/, Mh .s/, N0 .s/ such that deg M0 deg Mh
and deg M0 deg N0 . This transfer function
I
is proper
I
is analytic on C n , where is set of roots of M0 .s/ C Mh .s/e
I
has only poles as its singularities
To further simplify matters, assume also that there is
I
no pole / zero cancellations in G.s/
sh
D0
Asymptotic poles location
Let a ´ Pa .1/, where Pa ´
Theorem
Mh
M0
(ja j is the high-frequency gain of Pa ).
As jsj ! 1, poles of G.s/ are asymptotic to points with Re s D
lnja j
h
esh D Pa .s/ D a C O
1
s
sh
In this case, G.s/ has at most finitely many unstable poles. Moreover, the
following result can be formulated:
.
Lemma
D 0 or
Proof (outline).
Proof.
Poles are solutions of the characteristic equation M0 .s/ C Mh .s/e
ja j < 1
N 0.
Let ja j < 1. Then G 2 H 1 iff G.s/ has no poles in C
N 0 , it does not belong to H 1 .
If G.s/ has a pole in C
N 0 , there is a bounded S C
N 0 such that in C
N0nS
If G.s/ has no poles in C
1. M0 .s/ has no roots
ˇ N .s/ ˇ
0
ˇ < c for some ja j b < 1 and c > 0.
2. jPa .s/j < b and ˇ M
0 .s/
By Rouché’s arguments, as jsj ! 1 roots approach solutions of esh D a ,
i.e., sh D ln. a / C j2k , k 2 Z. Thus,
(
lnja j C j2k
if a < 0
sh !
lnja j C j.2k C 1/ if a > 0
Thus, jG.s/j is bounded in S (no poles and the set is bounded) and
which for a ¤ 0 approaches vertical line with Re s D lnja j and for a D 0
approaches Re s D 1.
Hence, G.s/ is analytic and bounded in C0 .
ja j > 1
ja j D 1: example 1
In this case, G.s/ has infinitely many unstable poles. Hence, the following
result can be formulated:
jG.s/j D
Let G.s/ D
I
jN0 .s/=M0 .s/j
c
<
< 1;
sh
1 b
j1 C Pa .s/e j
1
sC1C.sC2/e
s
N 0 n S:
8s 2 C
. This transfer function
has infinitely many poles in C0 , hence unstable.
Lemma
Let ja j > 1. Then G 62 H 1 .
Proof.
Obvious.
To see this, consider its characteristic equation at s D C j! :
s
C
j
!
C
2
. C 2/2 C ! 2
e e j! D
H) e D
C j! C 1
. C 1/2 C ! 2
Right-hand side here > 1 iff > 23 . Hence, whenever > 32 , e > 1 or,
equivalently, > 0. Since roots accumulate around D 0, there might be
only a finite number of roots in 32 .
ja j D 1: example 2
N 0 of poles of G.s/ satisfying
We know that there is a sequence fsk g 2 C n C
. This transfer function
N 0.
has no poles in C
Let G.s/ D
I
1
sC1Cs e
ja j D 1: example 2 (contd)
s
To see this, consider its characteristic equation at s D C j! :
ˇ
ˇ
ˇ ˇ
ˇ
ˇ
C j! C 1
1 ˇˇ ˇˇ
j!
ˇ
ˇ
e e
D
ˇ1 C 2
H) e D ˇ1 C
ˇ
2
C j!
C j!
C! ˇ
If D 0, 1 D 1 C 1=j!j: unsolvable. If > 0, e
1: contradiction.
sk C 1 C sk e
Let G.s/ D
I
G2H
1
.sC1/.sC1Cs e
1
s/
with limk!˙1 jsk j D 1 W
G. sk / D
1
1
sk
sk
e sk
D
1
1
D 1 C sk ;
sk C sk2 =.1 C sk /
so that jG. sk /j ! 1 C j2k C 1j . Thus, G.s/ unbounded on f sk g 2 C0 , so
G 62 H 1 and hence unstable.
ja j D 1: on the safe side
. In this case
Thus, we saw that in this case G.s/ might be stable. Yet this
I
(in fact, kG.s/k1 D 2). Hence,
I
D 0;
Then
I
ja j D 1: example 3
sk
stability is fragile (extremely non-robust).
Indeed,
G.s/ is stable.
I
infinitesimal increase of jPa .1/j leads to instability.
It is thus safe to regard such systems as practically unstable.
In general, if ja j D 1, the transfer function
N0 .s/
M0 .s/CM1 .s/e
sh
2 H 1 only if3
deg M0 .s/ deg N0 .s/ C 2:
3 For proof and further details see “H 1 and BIBO stabilization of delay systems of neutral
type,” by Partington and Bonnet, Systems & Control Letters, 52, pp. 283–288, 2004.
Internal stability and high-frequency gain
Summary
So far we learned that
wu
Pr .s/e
sh
I
I
C.s/
u
y
I
wy
This feedback system called internally stable if transfer matrix
1
1 Pr .s/C.s/e
sh
wy wu
7!
sh
µ
1
1 Lr .s/e
1
Pr .s/e sh
2 H1
C.s/ Pr .s/C.s/e sh
D
sh
ML .s/
ML .s/ NL .s/e
s
µ
N0 .s/
M0 .s/ M1 .s/e
I
s
;
this system cannot be internally stable.
Slight modification doing the trick
Theorem
Transfer function
G.s/ D
N0 .s/
M0 .s/ C Mh .s/e
sh
;
˚
deg M0 max deg Mh ; deg N0 ;
N ˛.
is (practically) stable iff 9˛ < 0 such that G.s/ has no poles in C
This result says that in time-delay systems (both retarded and neutral)
poles play essentially the same role as in the rational case,
N0 !C
N ˛ , for some ˛ < 0).
we just should slightly redefine stability region (C
This, in turn, makes it possible to
I
I
extend classical stability analysis methods to time-delay systems.
N 0 , yet make sure that
We still may use C
I
no pole chain is asymptotic to Re s D 0.
if ja j D 1, G.s/ practically unstable
Important point:
for which deg M0 .s/ D deg N0 .s/ (6 deg N0 .s/ C 2). Thus, if jLr .1/j D 1,
I
if ja j > 1, G 62 H 1 because it has infinitely many poles in C0
y
u ,
Its .1; 1/ entry, the sensitivity function, is of the form
1
1 Pr .s/C.s/e
N 0,
if ja j < 1, G 2 H 1 iff it has no poles in C
N 0 ” stability criterion might fail
classical “no poles in C