System identification
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14-02-2011, 03:26 PM

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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.
Other Definition

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.
System identification and parameter estimation
Model validation
Identification: time-domain & frequency-domain
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
• Open loop identification Car shock absorber testing
• Response loudspeaker
• Knee jerk reflex
• Etc, etc
Ti[b]me-domain & Frequency-domain
Cross-product function
Auto-product function
Covariance and correlation functions
Special cases of auto-covariance
Open loop identification(time domain)with cross-covariance
Open loop identification(frequen
cy dom[/b]ain)
Identification in the closed loop
• Time domain models for identification Least squares (LS)
• Output Error (OE)
Least squares model
• equation system-
• Least Squares (LS) Method (LUENBERGER, D. G. 1996)
• minimization =⇒the best linear nondeviated estimation
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 datalinear regression can be used for model parameters stimation
ARMAX model (Average model with eXternal input)enables us to model deterministic and stochastic parts of the system independentlylinear regression cannot be used for model parameters estimation→pseudolinear reg.
OE model (Output Error model)
Example - Model Identification Using ARX Model
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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