1. System identification
Course : MECHATRONICS Professor : Dr.A.GHANBARI Student : HOSSEIN NEJATBAKHSH Student Number :
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2. What is identification?
System identification is the art and science of building mathematical
models of dynamic systems from observed input-output data.
it’s an interface between the real world of applications and the mathematical world of control theory.
in control engineering the field of system identification uses statistical
methods to build mathematical models of dynamical systems from
measured data.
The core of estimating models is statistical theory.
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Other Definition
•Zadeh(1962)
Identification is the determination on the basis of input and
output, of a system within a class of systems, to which the
system under test is equivalent.
•Parameter estimation is the experimental
determination of values of parameters that govern the dynamic
and/or non-linear behaviour, assuming that the structure of the
model is known.
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4. System identification and parameter estimation
Input signal output signal
system identification
Input signal output signal
predicted output
Parameter estimation
unknown
system
unknown
system
model
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Model validation
Input output
validation
predected
unknown system
model
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Identification: time-domain & frequency-domain j
u(t),y(t)
U(ω),Y(ω)
non-parametric model
non-parametric model
Parametric model
Parametric
model
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7. Identification: time-domain & frequency-domain
U(t) y(t)
Time domain
• y(t) = ∫ h(t’)u(t-t’)*dt’ + n(t)
• Unknown system: impulse response of h(t’)
• Mostly: direct model parameterization
Frequency domain
• Y(ω) = H(ω)U(ω) + N(ω)
• Unknown system: transfer function H(ω) for number of
frequencies ?
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8. Open loop identification
n(t) U(t) y(t) • Car shock absorber testing • Response loudspeaker • Knee jerk reflex • Etc, etc ?
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9. Time-domain & Frequency-domain
Time Domain Fourier Frequency Domain
Transformation
input, output input, output
cross-product
Function
cross-covariance cross-spectral
function density
cross-correlation
Function coherence
x(t), y(t)
X(ω), Y(ω)
Rxy(τ)
Sxy(ω)
Cxy(τ)
Kxy(τ)
Γxy(ω)
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10. Cross-product function
x(t) T: observation time Y(t) τ: time shift parameter
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11. Auto-product function
X(t) T: observation time Y(t) τ: time shift parameter
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12. Covariance and correlation functions
Cross-covariance function: Auto-covariance function: Cross-correlation function: Auto-correlation function:
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13. Special cases of auto-covariance
Properties: (by definition) -White noise: - u(t) is white noise, with variance 1 - =0 for all T≠0
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14. Open loop identification(time domain) with cross-covariance
n(t) u(t) y(t) shift with τ : multiply with u(t): white noise : for , ?
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15. Open loop identification(frequency domain)
Auto-spectral densities
Alternative approach:
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16. Identification in the closed loop
Open loop: Open loop estimator used in closed loop:
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17. Time domain models for identification
Least squares (LS)
ARX
ARMAX
Output Error (OE)
FIR
And ...
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Least squares model
• equation system - • Least Squares (LS) Method (LUENBERGER, D. G. 1996) • minimization =⇒the best linear nondeviated estimation
vector of the left sides
m >> n data matrix exactly known
unknown vector
error – random variable
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Example
Using the Least Squares method approximate the points in Figure by a linear function.
We are looking for parameters k and q of equation
y = kx + q
%% parameters
k = 0.2; q = 1;
sigma2 = 0.5;
%% experiments
X = [ ]’;
y = k*X + q + ...
sigma2*randn(size(X));
Z = [X, ones(size(X))];
theta = inv(Z’*Z)*Z’*y
k = theta(1)
q = theta(2)
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20. Identification of Dynamical Systems
• ARX model (Auto Regresive model with eXternal input)
– prediction of mean value ˆy(t|t − 1) is linear function of measurable data
– linear regression can be used for model parameters estimation
• ARMAX model (Average model with eXternal input)
– enables us to model deterministic and stochastic parts of the system independently
– linear regression cannot be used for model parameters estimation→pseudolinear reg.
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21. OE model (Output Error model)
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22. Example - Model Identification Using ARX Model
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• difference equation • n-measuring with noise
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• compact form of the previous equation system
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Result of the identification experiment – step response
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FIR MODEL
Finite impulse response(FIR) models are frequently used in model
Predictive control (MPC) systems because they can fit arbitrarily
Complex stable linear dynamics.
However , identification of FIR models from experimental data my
result in data-over fitting and high modeling uncertainly.
To overcome this, FIR models may be determined by :
(a) regularization – based least squares, and
(b) indirectly after prior identification of other parametric models
such as ARX.
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References
System identification & parameter Estimation (SIPE) by A.schouten
Model identification by J.roubal ( czech Technical university)
Department of chemical Engineering Texas A&M university
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