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留学生数学建模论文Sampling and Quantization in Modelling and Identification
Graham C. Goodwin
Centre for Complex Dynamics Systems and Control
School of Electrical Engineering
Presented at Benelux Meeting on Systems and Control
March 13th - 15th, 2007
Topics to be covered include: sampled data models, coefficient quantization and delta models, sampling zero dynamics forlinear deterministic and stochastic systems, implications inrobust identification from sampled data, sampling zerodynamics for nonlinear systems, applications in systemidentification.
Content
I Interaction between Sampled Signals and AnalogueSystems
I Sampled Data Models for Linear Deterministic Systems
I Coefficient Quantization Issues
I Delta Operator
I Exploiting Connections between Continuous and Discrete
Time Models
I Re-evaluation
I Sampling Zeros for Linear Systems
I Asymptotic Sampling Zeros
I Robustness Issues in System Identification arising from
use of Sampled Data Models
I Sampled Data Models for Nonlinear Systems
I Conclusions
Question:
How do sampled signals interact with an analogue physical
system?
Two Issues:
(i) D/A conversion at input side (usually via some form of
hold)
(ii) A/D conversion at output side (usually including
anti-aliasing filtering)
The Elements of the Sampling Process
Sample
yk
Continuous-time System
Differential Equations
Physical knowledge:
˙v
(t)
Hold
uk set of (nonlinear)
u(t) y(t)
Sampled-data model
I Sampled-data models depend upon:
I The physical system,
I The hold device (ZOH, FOH, . . . )
I The sampling device (anti-aliasing filters, . . . )
Content
I Interaction between Sampled Signals and Analogue
Systems
I Sampled Data Models for Linear Deterministic
Systems
I Coefficient Quantization Issues
I Delta Operator
I Exploiting Connections between Continuous and Discrete
Time Models
I Re-evaluation
I Sampling Zeros for Linear Systems
I Asymptotic Sampling Zeros
I Robustness Issues in System Identification arising from
use of Sampled Data Models
I Sampled Data Models for Nonlinear Systems
I Conclusions
Deterministic Linear Systems
Once the hold and samples have been specified it is easy to
obtain sampled data models for linear case.
I Continuous-time description:
d
dt x(t) = A x(t) + Bu(t)
y(t) = C x(t)
Y(s) = C(s In − A)−1B | {z }
G(s)
U(s)
I Discrete-time model:
xk+1 = Aq xk + Bquk
yk = Cq xk
Y(z) = Cq(z In − Aq)−1 | {z B}q
Gq(z)
U(z)
8< :
Aq = eA
Bq =
R
0 eAB d
Cq = C
and
#p#分页标题#e#
u(t) = uk
t2[k,k+)
ZOH input
This SD model is exact, i.e.:
yk = y(k)
Write in terms of the shift operator
qxk = Aqxk + Bquk
yk = Cqxk
where ‘q’ is the forward shift operator
qxk = xk+1
Content
I Interaction between Sampled Signals and Analogue
Systems
I Sampled Data Models for Linear Deterministic Systems
I Coefficient Quantization Issues
I Delta Operator
I Exploiting Connections between Continuous and Discrete
Time Models
I Re-evaluation
I Sampling Zeros for Linear Systems
I Asymptotic Sampling Zeros
I Robustness Issues in System Identification arising from
use of Sampled Data Models
I Sampled Data Models for Nonlinear Systems
I Conclusions
A major difficulty with shift operator models is that, with fast
sampling, a near perfect model is
yk+1 = yk
Thus, if we consider a model such as
yk+1 = ayk + buk
Then the behaviour depends on the difference of function from
1!
More generally, if we consider
yk+n = an−1yk+n−1 + . . . + a0yk = buk
Then behaviour depends on difference of coefficients from the
Binomial Coefficients.
Illustration
Consider the following 2 Models
(a)
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