1 lecture 2 sampling of systems graham c. goodwin by centre for complex dynamic systems and control...
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1
Lecture 2
Sampling of Systems
Graham C. Goodwinby
Centre for Complex Dynamic Systems and ControlUniversity of Newcastle, Australia
Presented at the “Zaborszky Distinguished Lecture Series”December 3rd, 4th and 5th, 2007
2
Overview
• High performance signal processing and control depends, inter-alia, on the availability of accurate models to represent systems
• Underlying physical system typically continuous• Modern data recording equipment inevitably uses
some form of sampling and quantization• This raises the question of the relationship
between the sampled data and the underlying continuous time system
• We will study this question for linear and nonlinear systems
3
Outline
1. The Elements of Sampling2. Sampled Data Models for Linear Deterministic Systems (A first look)3. Shift Operator4. Delta Operator5. Some advantages of delta models6. Mid course correction; beyond Euler integration7. More accurate sampled data models for deterministic linear systems8. Sampled data models for stochastic linear systems9. CAR estimation revisited10. Sampled data models for deterministic nonlinear systems11. Sampled data models for stochastic nonlinear systems12. Conclusions
4
Question
How do sampled signals interact with an analogue physical system?
Two Issues:
i. D/A conversion at input side (via some form of hold)
ii. A/D conversion at output side (including anti-aliasing filtering)
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The Elements
PhysicalSystem
Hold PresampleFilter
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Outline
1. The Elements of Sampling2. Sampled Data Models for Linear Deterministic Systems (A first look)3. Shift Operator4. Delta Operator5. Some advantages of delta models6. Mid course correction ; beyond Euler integration7. More accurate sampled data models for deterministic linear systems8. Sampled data models for stochastic linear systems9. CAR estimation revisited10. Sampled data models for deterministic nonlinear systems11. Sampled data models for stochastic nonlinear systems12. Conclusions
7
Deterministic Linear Systems
Once the hold and presample filter have been specified it is easy to obtain exact sampled data models for linear case.
ZOH input
• Continuous-time description:
• Discrete-time model:
[ )( )
,
u t ukt k k
=
Î D D +D
( ) ( ) ( )
( ) ( )
dx t Ax t Bu t
dty t Cx t
= +
=
1x A x B uk q k q k
y C xk q k
= ++
=
0
AA eq
AB e Bdq
C Cq
h h
ì Dï =ïïïï Dïï =íïïïï =ïïïî
ò
8
Outline
1. The Elements of Sampling2. Sampled Data Models for Linear Deterministic Systems (A first look)3. Shift Operator4. Delta Operator5. Some advantages of delta models6. Mid course correction ; beyond Euler integration7. More accurate sampled data models for deterministic linear systems8. Sampled data models for stochastic linear systems9. CAR estimation revisited10. Sampled data models for deterministic nonlinear systems11. Sampled data models for stochastic nonlinear systems12. Conclusions
9
Typically, we write the above discrete model in terms of the shift operator as
where ‘q’ is the forward shift operator
k q k q k
k q k
qx A x B u
y C x
= +
=
1k kqx x +=
10
However, a difficulty with shift operator models is that, with fast sampling, we almost have
Thus, if we consider a model such as
Then, we might anticipate that a 1; b 0 as 0.More generally, if we consider an nth order AR description:
Then the “a coefficients” tend to the Binomial Coefficients.
1k kx x+ =
1k k kx ax bu+ = +
1 1 0k n n k n k ky a y a y bu+ - + -= + + =K
11
Question
What happens with coefficient quantization (i.e. finite word length representations)?
Consider the following 2 models
Note that the coefficients differ by 1%.
Hence if the coefficients were to be quantized (say to 6 bits) then the models would be identical!
Question: Does this really matter?Maybe the systems are very similar.
( ) [ ]
( ) [ ]
2
2
(a)
(
1.9 0.9025 0
1.9 0.8925 0b)
q q y k
q q y k
- + =
- + =
12
Surprising Fact
Coefficients differ by 1% yet (a) is stable (b) is unstable.
( ) [ ]
( ) [ ]
2
2
(a)
(
1.9 0.9025 0
1.9 0.8925 0b)
q q y k
q q y k
- + =
- + =
13
• How can we better represent the system?• Idea: Instead of modelling the absolute
displacement, i.e.,
• How about we model the difference
• This is the core idea in delta domain models.
( )1k ky f y+ =
( )1 'k k ky y f y+ - =
14
Outline
1. The Elements of Sampling2. Sampled Data Models for Linear Deterministic Systems (A first look)3. Shift Operator4. Delta Operator5. Some advantages of delta models6. Mid course correction ; beyond Euler integration7. More accurate sampled data models for deterministic linear systems8. Sampled data models for stochastic linear systems9. CAR estimation revisited10. Sampled data models for deterministic nonlinear systems11. Sampled data models for stochastic nonlinear systems12. Conclusions
15
Delta Operator
Core attributes• This centers the calculation by subtracting what is
already known• The same idea is used in many areas:
– Sigma Delta Modulator– Predictive Coding– Optimal Noise Shaping Quantization etc
(See later lectures in this series)
( ) ( )( )
x t x tx td
é ù+D -ê úê úDë û
@
16
Coefficient Quantization Revisted• Recall shift operator example:
• Coefficients differ by 1% yet (a) is stable (b) is unstable.• Equivalent delta operator models (assuming = 0.1)
• Coefficients differ by 400%. Stability obvious by analogy with continuous time.
( ) [ ]
( ) [ ]
2
2
(a) 1.9 0.9025 0
(b) 1.9 0.8925 0
q q y k
q q y k
- + =
- + =
( ) [ ]
( ) [ ]
2
2
(c) 0.25 0
(d) 0.75 0
y k
y k
d d
d d
+ + =
+ - =
17
Some History
• 17th Century Numerical Analysis.
• Harriot (teacher of Sir Walter Raleigh) developed accurate interpolation formulae based on finite differences.
• Newton and Stirling in 18th Century developed formal calculus of differences.
• Lagrange and Laplace studied relationships between linear difference equations in shift and delta form.
• Cauchy (1827) developed operational calculus for finite differences.
• Finite difference central to numerical analysis and signal processing.
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Many Modern Contributors
• Tschauner (1963)• Halkin (1966) – The Discrete Maximum Principle• Jury (1971) – Unified Stability Theory• Agarwal and Burrus (1975) – Signal Processing• Kitamon (1983) – Unified Control Theory• Orlandi and Martinelli (1984) – Signal Processing• Goodall and Brown (1985) – Microprocessor Implementation• Middleton and Goodwin (1986) Filtering and Control• Goodwin, Gevers, Mayne, Middleton (1988) – Unified Theory
of Stochastic Adaptive Control• Weller, Feuer, Goodwin, Poor (1993) – Unified Lattice
Filtering
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State Space EquivalenceThe discrete state space model can be expressedin Delta form as
k k k
k k k
x t x t u t
y t x t u t
A B
C D
where , andq q C C C D D D
2 2
0
12! 3!
e I
e d I
A
A
A
B ΩB
A AΩ
20
Recall shift model
1
0
( )
I
x t e x t e I u tA AA B
Thus, we recover the continuous model as ( 0).
Also, discrete results expressed in form continuous as ( 0).
1e I x t e I u tx t x tA AA B
A
B
21
Outline
1. The Elements of Sampling2. Sampled Data Models for Linear Deterministic Systems (A first look)3. Shift Operator4. Delta Operator5. Some advantages of delta models6. Mid course correction ; beyond Euler integration7. More accurate sampled data models for deterministic linear systems8. Sampled data models for stochastic linear systems9. CAR estimation revisited10. Sampled data models for deterministic nonlinear systems11. Sampled data models for stochastic nonlinear systems12. Conclusions
22
Some Advantages of Models
i. Unification
ii. Numerical Properties
iii. Insight
23
(i) Unification
• Many (arguably all) results in the literature exhibit a close connection between discrete and continuous:
– Checking discrete stability – Jury type testing in delta domain converges to Routh-Hurwitz as 0. (Jury, Premaratne, …)
– Discrete-time Riccati equations for H2/H optimization converge to continuous-time Riccati equations as 0. (Middleton, Salgado, …)
– Constrained linear-quadratic optimal control. (Feuer, Yuz)
24
Example 1: Discrete Delta TransformWe define the Discrete Delta Transform pair as:
The Discrete Delta Transform can be related to Z-transform
by noting that
where . Conversely
0
11
1
11
2
k
k
k
D y k Y y k
D Y y k Y dj
1( )q z
Y Y z
11
( ) zqY z Y
( )qY z Z k
25
Discrete Delta Transform
26
Example 2: Stability
Z domain
1
domain
x Ax Bu x x
1k k kx Ax Bu 1k kqx x
1 ' 'k kk k
x xA x B u
1k kk
x xx
1
27
Example 3
Continuous Riccati Discrete Riccati
T T
TI
APA CPC
A CPC1
T T
T
P Ω PA AP H Γ H
H PC S Γ
28
(ii) Numerical Properties
Output
Time
Comparison of step response of continuous-time systems to the step responses of discrete-time approximation systems using shift and operator implementations of limited word length.
29
It’s even worse than this suggests!
Actually 3 more significant digits were used in the shift model than in the model!
30
Comparison of errors for dynamic Riccati equation using delta ( ) and shift (q) operators.
Solving Discrete Riccati Equation
31
(iii) Insights – Generally all results expressed in terms of delta operator converge to the
continuous time result as 0. Example: State Space Models expressed using -operator
rather than q-operator• Rewriting in terms of -operator:
Then, as 0
i.e., the underlying CT system representation is recovered.
1qd
-=
D
k k k
k k
x A x B u
y Cxd ddì = +ïïíï =ïî
0
where1
An
A
e IA
B e Bd
d
hd h
D
D
ìï -ï =ïï Dïíïï =ïï Dïî ò
dA A B B
t d dd® ® ®
32
Outline
1. The Elements of Sampling2. Sampled Data Models for Linear Deterministic Systems (A first look)3. Shift Operator4. Delta Operator5. Some advantages of delta models6. Mid course correction ; beyond Euler integration7. More accurate sampled data models for deterministic linear systems8. Sampled data models for stochastic linear systems9. CAR estimation revisited10. Sampled data models for deterministic nonlinear systems11. Sampled data models for stochastic nonlinear systems12. Conclusions
33
• The ideas presented so far have emphasized the close connections between discrete and continuous cases.
• Indeed, simply replacing (continuous) by (discrete) yields a discrete time model of accuracy . [This corresponds exactly to the use of Euler Integration].
• Is this always adequate?
d
dt1qd
-=
D( )O D
34
A Motivational Problem• Continuous-time Auto-Regressive (CAR) identification:
where : continuous-time “white noise”• It is tempting to think that, for 0, we could think of
the sampled-data model as being simply
• With : discrete-time “white noise”• Then we could use ordinary least squares to estimate
v&
kw
1
1 01
n n
nn n
d y d ya a y v
dt dt
-
- -+ + + = &K
11 0
n nk n k k ky a y a y wd d -
-+ + + =K
[ ]1 2 0, , ,n na a aq - -= K
35
• However, this does not happen. Consider:
• “Euler” integration gives:
• However, L.S. estimation using the above approximate sampled-data model gives
0
1 1
2ˆ
3a a
D®
®
2
1 02
d y dya a y v
dt dt+ + = &
21 0ˆ ˆk k k ky a y a y wd d+ + =
36
• What has gone wrong?It turns our that we need a more accurate representation than Euler which we recall had error O().
• The source of the difficulty is not the use of the delta operator, per se, but a problem due to relative degree as we will see below.
• We will next find alternative sampled data models that are essentially as easy to obtain as these from Euler integration but which have error O(r) where r is the relative degree!
37
Outline
1. The Elements of Sampling2. Sampled Data Models for Linear Deterministic Systems (A first look)3. Shift Operator4. Delta Operator5. Some advantages of delta models6. Mid course correction ; beyond Euler integration7. More accurate sampled data models for deterministic linear systems8. Sampled data models for stochastic linear systems9. CAR estimation revisited10. Sampled data models for deterministic nonlinear systems11. Sampled data models for stochastic nonlinear systems12. Conclusions
38
To motivate the subsequent analysis, consider simple case of double integrator
• Continuous state space model is
2
2
d yu
dt=
1 1
2 2
0 1 0
0 0 1
x xdu
x xdt
é ù é ùé ù éùê ú ê úê ú êú= +ê ú ê úê ú êúë û ëûë û ë û
39
Say that u is generated by a ZOH and we sample output at interval .
• Associated discrete time model
[ ]
21 12 2
2 21
2
1
2
1
2
(exact)
1
2 2
12
0 1
1 0
k k
k
k
k
x x AI A A Bu
x x
xu
x
xy
x
+
ì üé ù é ùì ü ï ïDï ïï ï ï ïê ú ê ú= + D + D + D +í ý í ýê ú ê úï ï ï ïï ïî þë û ë û ï ïî þé ùDé ùé ùD ê úê úê ú= +ê úê úê ú ê úë ûë û Dë û
é ùê ú=ê úë û
K K
40
Hence exact sampled model has transfer function:
Note that the discrete time model has additional zero dynamics (which are sometimes called “sampling zeros”).
( )
[ ]
( )
1
2
2
12
1
q q q qT z C ZI A B
z
z
-é ù= -ë ûD
+=
-
( ) 2
12Gd
g
gg
D+
=
or in form:
surprise?
41
More Generally
where Pr denotes the “sampling zero dynamics”.
( )1 rr r
P
s
d
d
D®
42
Sampling Zeros with Zero Order Hold on Input
Relative Degree
2
3
rP
11
2
211
6
43
• In general, the sampling zero dynamics are given by the Euler-Fröbenius polynomials (Åström, Hagander and Sternby, 1984)
• In -operator form (Yuz and Goodwin, 2005):
where
( )( )r
r
PGd
gg
g
D=
( ) ( )
1
2
12! !
1det1 !
0 1
r
r
r
r
pr
gg
g
-
-
é ùD Dê úê úê úê úDê ú-D = ê ú-ê úê úê úê ú-ë û
L
L
M O O M
L
This will reappear later
44
Generalization to arbitrary transfer functions
At high frequencies this model behaves like
Hence expect
Indeed, it can be shown that this approximate model has local truncation error O(r).
(Compare with O() for Euler integration.)
( )( )( )r
F sG s
E s=
( )m
m mc n r
b s bG s
s s=;
( )( )
( )F
GEd
gg
g;
Relative degree is important!
degree relative degree
degree
mr n m
n= -
( )( ) ( )
( )rPF
GEd
gg
g
gD;
45
Summary
• More accurate models are obtained by going beyond Euler integration.
• These more accurate models have an intuitive representation in delta form.
• We thus advocate– the use of higher order integration which, inter
alia, leads to sampling zero dynamics, and– the use of the delta operator.
46
Outline
1. The Elements of Sampling2. Sampled Data Models for Linear Deterministic Systems (A first look)3. Shift Operator4. Delta Operator5. Some advantages of delta models6. Mid course correction ; beyond Euler integration7. More accurate sampled data models for deterministic linear systems8. Sampled data models for stochastic linear systems9. CAR estimation revisited10. Sampled data models for deterministic nonlinear systems11. Sampled data models for stochastic nonlinear systems12. Conclusions
47
Similar Ideas apply to Stochastic Linear Systems
• Stochastic system or noise models:
where v(t) is a Wiener process:
• Exact discrete-time model:
( ){ }2 2vE dv t dts=
1
k q k k
k q k
x A x V
y C x
+ìï = +ïíï =ïî
%
( ) ( ) ( )( ) ( ) ( )( ) ( )
dx t Ax t Bdv ty t H v t
y t Cx tr
ì = +ïï= íï =ïî&
{ } [ ]
2
0
Discrete white noise: he e
w r
Aq
k
Tk q K
TA T Aq v
A e
V
E V V k
e BB e dh h
d
s h
D
D
ìï =ïïïïïïí =W -ïïïïï W =ïïî ò
l
%
%% l
48
There exists a parallel development for stochastic sampling zero dynamics. Here the anti-aliasing filter plays a dual role to the ZOH for the input signal.
The dual of a ZOH is an integrate and sample filter.
( ) ( )1 t
ty t y dt t
- D=
D ò( )
( )( ) ( )( ) ( )
*
*continu ous
k
y
y y k
B s B ss
A s A s
= D
F =
( )( ) ( ) ( )
( ) ( )
*
*2 discrete y
rB B
A A
Pdg
gd
d dF Þ( )
( ) ( )( ) ( )
*
*discrete y
B B
A A
d dg
d dF Þ
49
Outline
1. The Elements of Sampling2. Sampled Data Models for Linear Deterministic Systems (A first look)3. Shift Operator4. Delta Operator5. Some advantages of delta models6. Mid course correction ; beyond Euler integration7. More accurate sampled data models for deterministic linear systems8. Sampled data models for stochastic linear systems9. CAR estimation revisited10. Sampled data models for deterministic nonlinear systems11. Sampled data models for stochastic nonlinear systems12. Conclusions
50
CAR Identification Revisited
• Recall
but LS gave
• Solutions:1. Use asymptotic sampling zeros
2
1 02
d y dya a y v
dt dt+ = &
0
1 1
2ˆ
3a a
D®
®
21 0
1ˆ ˆ 1
3 3k k k ky a y a y wd d d d
æ ö÷ç+ + = + D ÷ç ÷çè ø-
21 0ˆ ˆ? k k k ky a y a y wd d dÞ + + =
Spectral factor of ( )2P dD
51
Implications:
• Should use Filtered Least Squares
• Then and
( )( )
2
21 01
22
1J y a y a y
Pd d
d
é ùê ú= + +ê úê úDë û
å
1 1a a® 0 0a a®
52
Outline
1. The Elements of Sampling2. Sampled Data Models for Linear Deterministic Systems (A first look)3. Shift Operator4. Delta Operator5. Some advantages of delta models6. Mid course correction ; beyond Euler integration7. More accurate sampled data models for deterministic linear systems8. Sampled data models for stochastic linear systems9. CAR estimation revisited10. Sampled data models for deterministic nonlinear systems11. Sampled data models for stochastic nonlinear systems12. Conclusions
53
• Continuous-time nonlinear systems are described by differential equations.
• In general, exact sampled-data models are impossible to obtain for nonlinear case.
• However, approximate discrete-time descriptions can be obtain by numerical integration (analogously to linear case).
54
• We consider a class of deterministic nonlinear systems
• We assume that the system has nonlinear relative degree r i.e.,
( ) ( )( ) ( )( ) ( )
( ) ( )( )x t f x t g x t u t
y t h x t
= +
=
&
( ) ( )
( )10
0 0, ,for ani
(ii)
2, d
0
kg f
rg f
L L h x k r
L L h x-
= = -
¹
K
55
• We express the system in normal form:
( ) ( )( )
1 2
1
, ,
,
r r
r
z z
z z
z b a u
q
Vh Vh
h Vh
-
=
=
= +
=
&
M&
&
&
[ ]1 2
1 2
, , ,
, , ,
T
r
T
r r n
z z z
z z z
V
h + +
ìï =ïïíï é ù=ï ë ûïî
& K
Kwhere y = h(x) = z1 and
56
Core Insight
• The output y = z1 has a chain of r integrators (as in the linear case). Hence one might wonder if one carries out the numerical integration in an appropriate fashion, then we may find additional discrete zero dynamics as in the linear case.
• Actually, this turns out to be true. However, we need to change the order of the Taylor series expansion depending upon the “depth” from y.
57
• Thus we perform Taylor series expansion for each state as follows:
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
2
1 1 2 3
1
2 2 3
, ,2 !
, ,1 !
, ,
,
r
t k
r
t k
r r t k
t k
z k z k z k z k b a ur
z k z k z k b a ur
z k z k b a u
k k qx
Vh Vh
Vh Vh
Vh Vh
h h Vh
= D
-
= D
= D
= D
D D é ùD +D = D +D D + D + + +ë û
D é ùD +D = D +D D + + +ë û-
é ùD +D = D +D +ë ûé ùD +D = D +D ë û
K
K
M
58
• Converting to -form we obtain the following approximate DT model:
( )
( ) ( )
( ) ( )( )
( )
2 1
3 2
0 12 1 ! !
0 0 12 ! 1 !
, ,
0 0 12
0 0 0 1
,
r r
r r
S S S S S S
S S S
r r
r r
b a u
q
dV V V h V h
dh V h
- -
- -
é ù é ùD D Dê ú ê úê ú ê ú-ê ú ê úê ú ê úD Dê ú ê úê ú ê ú- -ê ú ê úê ú ê úê ú ê úê ú ê ú= + +ê ú ê úê ú ê úê ú ê úDê ú ê úê ú ê úê ú ê úê ú ê úê ú ê úê ú ê úê ú ê úë û ë û
=
L
L
M O O M M
L
L
59
Accuracy
• The local truncation error in the output is of order , i.e.:1r+D
( ) ( ) 1 for some c onstant S ry k y k CC +D - D < ×D
60
Zero Dynamics
• The DT nonlinear zero dynamics are given by:
– where
– A linear subsystem of dimension r – 1:
( )2:0, ,S S Srq zdh h= %
2: 2 , ,TS S S
r rz z zé ù=ê úë û% % %K
2: 22 2:S S
r rz Q zd =% %
61
• Moreover, the eigenvalues of the linear subsystem are given by
This is exactly as in the linear case!
That is we have the same (asymptotic) sampling zero dynamicsas for a linear system of the same relative degree r
( ) ( )1 22 1
!det r rr
rI Q pg g- -- = D
D
( ) ( )
1
2
12! !
1det1 !
0 1
r
r
r
r
pr
gg
g
-
-
é ùD Dê úê úê úê úDê ú-D = ê ú-ê úê úê úê ú-ë û
L
L
M O O M
L
62
Outline
1. The Elements of Sampling2. Sampled Data Models for Linear Deterministic Systems (A first look)3. Shift Operator4. Delta Operator5. Some advantages of delta models6. Mid course correction ; beyond Euler integration7. More accurate sampled data models for deterministic linear systems8. Sampled data models for stochastic linear systems9. CAR estimation revisited10. Sampled data models for deterministic nonlinear systems11. Sampled data models for stochastic nonlinear systems12. Conclusions
63
• We consider a special class of stochastic (nonlinear) systems:
where dvt are increments of a Wiener process• The diffusion team integral cannot be
interpreted in the usual Riemann-Stieljes sense.
( ) ( ) ( )
( ) ( )00 0
, , ; ,
, ,
t t t t t t
t t
t
dx a t x dt b t x dv y g t x
x x a x d b x dvt t tt t t
= + =
= + +ò ò
64
• To perform Taylor’s series-like expansions we need to consider the Ito rule of stochastic calculus: Given
and yt = g(t, xt), the usual chain rule has to be modified:
• We also apply the Ito rule to the process:
• Doing this, we obtain Ito-Taylor series expansion:
where depends on double stochastic integrals2sR
( ) ( ), ,t t t tdx a t x dt b t x dv= +
( )2
2
2
1
2
g g gdy dt dx dx
t x x
¶ ¶ ¶= + +
¶ ¶ ¶
( ) ( )00 0
, ,t t
tx x a x d b x dvt t tt t t= + +ò ò
( ) ( )0 0 0 20 0
t ts
tx x a x d b x dv Rtt= + + +ò ò
65
Different types of integration lead to discrete models of different accuracy; Specifically we will use the following:
• Strong convergence of order > 0, if there exists a constant C > 0 and a sampling period 0 > 0 such that:
• The simplest integration is the Euler-Maruyama approximation:
where and:
• This approximation contains time and Wiener integrals of multiplicity 1 only. Thus, it can be interpreted as an order 0.5 strong Ito-Taylor approximation.
( ){ } 0;kE x k x C gDD - £ D "D <D
( )kx x k= D
( ) ( )1k k k k kx x a x b x v+ = + D + D
{ }
( ){ }2
0kk
kk
k
E vv dv
E vt
D+D
D
ì D =ïïïD = íï D =Dïïîò
66
• We consider a class of nonlinear stochastic systems expressed in normal form as:
( ) ( )
1 2
1r r
r t
dy dx x dt
dx x dt
dx a X dt b X dv-
= =
=
= +
M
A class of nonlinear stochastic systems having relative degree n
67
• Expanding using Taylor series of different orders, we obtain the following approximate discrete time model:
( )( ) ( )
1
1 2
1 !1 !
0
12
0 0 1
rr
k k k k k
rr
X X a X b X V
-
+
é ùDé ù ê úDê úD ê úê ú- ê úê ú ê úê ú ê ú= + +ê ú ê úDê ú ê úDê ú ê úê ú ê úê úë û Dê úë û
L
M %O O M
M O
L
Theorem:
The output converges strongly to the true output with order r/2.
( ) ( )1y k x kD = D
68
Some Observations
• This discrete time model has a vector noise output of dimension r.
• In the linear case, we used spectral factorization (superposition) to reduce this to a scalar input noise source.
• In the nonlinear case, we cannot use superposition.• Nonetheless, it is possible to combine the
stochastic inputs by mimicking the linear case.• Then, we find that the resultant model has
stochastic sampling zero dynamics which parallel those for the linear case.
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Outline
1. The Elements of Sampling2. Sampled Data Models for Linear Deterministic Systems (A first look)3. Shift Operator4. Delta Operator5. Some advantages of delta models6. Mid course correction ; beyond Euler integration7. More accurate sampled data models for deterministic linear systems8. Sampled data models for stochastic linear systems9. CAR estimation revisited10. Sampled data models for deterministic nonlinear systems11. Sampled data models for stochastic nonlinear systems12. Conclusions
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Conclusions
• We have studied sampled data models for continuous time systems.• We have argued that coefficient quantization can be an issue with
sampled data models expressed in terms of shift operator.• Usually preferable to use Delta operator.• Delta operator provides unification with continuous time results,
improved numerical properties and insight.• Sampled data models generally contain extra sampling zero dynamics.• Sampling zero dynamics can be easily (asymptotically) characterized
in linear case.• Use of asymptotic zero dynamics leads to a model of accuracy O(r).• Ideas can be extended to classes of nonlinear deterministic and
stochastic systems.
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References• Interaction between Sampled Signal and Analogue Systems
– R.H. Middleton and G.C. Goodwin, Digital Control and Estimation, Prentice Hall, Englewood Cliffs, 1990.
• Sampled Data Models for Linear Deterministic Systems– A. Feuer and G.C. Goodwin, Sampling in Digital Signal Processing and control,
Birkhäuser, Boston, 1996.• Coefficient Quantization Issues
– R.H. Middleton and G.C. Goodwin, ‘Improved finite word length characteristics in digital control using delta operators’, IEEE Transactions on Automatic Control, Vol.31, No.11, pp.1015-1021,1986.
• Delta Operator– R.M. Goodall and Donoghue, ‘Very high sample rate digital filters using the
delta operator’, Proceedings of IEE, G, Vol.140, No.3, pp.199, 1993.– G.C. Goodwin, R.H. Middleton and H.V. Poor, ‘High-speed digital signal
processing and control’, Proceedings of IEEE, Vol.80, No.2, pp.240-259, 1992.– M.J. Newman And D.G. Holmes, ‘Delta operator digital filter for high
performance inverter applications’, IEEE Transactions on Power Electronics, Vol.18,m No.1, Part2, 2003.
– K. Premaratne and E.I. Jury, ‘Tabular method for determining root distribution of delta operator formulated real polynomials’, IEEE Transactions on Automatic Control, Vol.39, No.2., pp.352-355, 1994.
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References• Delta Operator Cont.
– K. Premaratne and E.I. Jury, ‘Application of polynomial array methods to discrete time system stability’, Proceedings of IEE, D, Vol.140, No.3, pp.198-204, 1993.
– K. Premaratne, S. Touset and E.I. Jury, ‘Root distribution of delta operator formulated polynomials’, Proceedings of IEE, D, Vol.147, No.1, pp.1-12, 2000.
– M.E. Salgado, R.H. Middleton and G.C. Goodwin, ‘Connections between continuous and discrete Riccati equation with applications to Kalman filtering’, Proceedings of IEE, D, Vol.135, pp.28-34, 1988.
• Exploiting Connections between Continuous and Discrete Time Models– H.H. Fan, ‘Efficient zero location tests for delta operator based polynomials’, IEEE
Transactions on automatic Control, Vol.42, No.5, pp.722-727, 1997.• Re-evaluation
– E.K. Larsson and T. Söderström, ‘Continuous-time AR parameter estimation using properties of sampled systems’, Proceedings of IFC World congress, Barcelona, 2002.
• Sampling Zeros for Linear Systems– K.J. Åström, P. Hagander and J. Sternby, ‘Zeros of sampled systems’, Automatica,
Vol.29, No.1, pp.31-38, 1984.
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References
• Asymptotic Sampling Zeros– B. Wahlberg, ‘Limit results for sampled systems’, International Journal of Control,
Vol.48, No.3, pp.1267-1283, 1988.– S. Weller, W. Moran, B. Ninness and A. Pollington, ‘Sampling zeros and the Euler-
Fröbenius polynomials’, IEEE Transactions on Automatic Control, Vol.46, No.2, pp.340-343, 2001.
• Robustness Issues in System Identification arising from use of Sampled Data Models
– G.C. Goodwin, J.I. Yuz and H. Garnier, ‘Robustness issues in continuous-time system identification from sampled data’, in Proceedings of 16 th IFAC World Congress, Prague, Czech Republic, 2005.
– T. Söderström, H. Fan, B. Carlsson and S. Bigi, ‘Least squares parameter estimation of continuous-time ARX models from discrete-time data’, IEEE Transactions on Automatic Control, Vol.42, No.5, pp.959-973, 1997.
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References
• Sampled Data Models for Nonlinear Systems– J.I. Yuz and G.C. Goodwin, ‘On sampled data models for nonlinear systems’, IEEE
Transactions on Automatic Control, Vol.50, No.10, pp.1477-1489, 2005.– J.I. Yuz and G.C. Goodwin, ‘Sampled data models for nonlnear stochastic systems’,
IFAC Symposium on system Identification, Newcastle, Australia, 2006.– S. Monaco and D. Normand-Cyrot, ‘Zero dynamics of aampled nonlinear systems’,
Systems and control Letters, Vol.11, pp.229-234, 1988.– S. Monaco and C. Normand-Cyrot, ‘A unified representation for nonlinear discrete-
time and sampled dynamics’, Journal of Mathematics Systems, Estimation and Control, Vol.7, No.4, pp.477-503, 1997.
– N. Kazantzis and C. Krvaris, ‘System-theoretic properties of sampled-data representations of nonlinear systems obtained via taylor-lie series’, International Journal of Control, Vol.67, No.9, pp.997-1020.1997.
– G.C. Goodwin, J.I. Yuz and M.E. Salgado, ‘Insights into the zero dynamics of sampled-data models for linear and nonlinear stochastic systems’, European Control Conference, Greece, 2-5 July, 2007.
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Lecture 2
Sampling of Systems
Graham C. GoodwinUniversity of Newcastle
Australia
by
Centre for Complex Dynamic Systems and ControlUniversity of Newcastle, Australia
Presented at the “Zaborsky Distinguished Lecture Series”December 3rd, 4th and 5th, 2007