PROCESS MODELLING AND MODEL ANALYSIS © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Statistical Model Calibration and Validation C12

PROCESS MODELLING AND MODEL ANALYSIS 2 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Overview   Grey box models and model calibration   Data analysis and preprocessing   Model parameter and structure estimation:   linear-nonlinear   static-dynamic   Model validation

PROCESS MODELLING AND MODEL ANALYSIS 3 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences A Systematic Modelling Procedure Problem definition Controlling factors Problem data Model construction Model solution Model verification Model calibration & validation

PROCESS MODELLING AND MODEL ANALYSIS 4 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Grey-box Models Process models   developed from first engineering principles (white box part)   part of their parameters and/or structure unknown (black box part) are called grey-box models

PROCESS MODELLING AND MODEL ANALYSIS 5 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Model Calibration Conceptual Problem Statement Given:  grey-box model  calibration data (measured data)  measure of fit (loss function)  Estimate: the parameter values and/or structural elements

PROCESS MODELLING AND MODEL ANALYSIS 6 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Model Calibration Conceptual Steps of Solution  Analysis of model specification  Sampling of continuous time dynamic models  Data analysis and preprocessing  Model parameter and structure estimation  Evaluation of the quality of the estimate

PROCESS MODELLING AND MODEL ANALYSIS 7 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Sampling of Continuous Time Dynamic Models Equi-distant zero-order hold sampling Discrete time input signal: u : {u(k)=u(t k ) | k=1,2,...} output signal: y : {y(k)=y(t k ) | k=1,2,...}

PROCESS MODELLING AND MODEL ANALYSIS 8 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Sampling of Continuous Time Dynamic Models ) Model parameters (1st order approximation ) Discrete time model equations: model parameters:  = I+Ah,  = Bh ( , ,C,D) Continuous time model equations: model parameters: (A,B,C,D)

PROCESS MODELLING AND MODEL ANALYSIS 9 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Data Analysis and Preprocessing -Data Screening-   gross errors   outliers   trends  Data visualization  Outlier tests  Trends, steady state tests  Gross error detection Check measured data for: Methods to be used include:

PROCESS MODELLING AND MODEL ANALYSIS 10 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Data Screening - Visualization Gross errors

PROCESS MODELLING AND MODEL ANALYSIS 11 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Data Screening - Visualization Trends and jumps

PROCESS MODELLING AND MODEL ANALYSIS 12 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Experimental Design for Parameter Estimation Static models   number of measurements   test point spacing   test point sequencing Dynamic models (in addition)   sampling time selection   excitation Pseudo Random Binary Signal (PRBS)

PROCESS MODELLING AND MODEL ANALYSIS 13 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Model Parameter and Structure Estimation   Conceptual problem statement   Least Squares parameter estimation - estimation procedure - properties of the estimate - linear and nonlinear models   Parameter estimation for static models   Parameter estimation for dynamic models

PROCESS MODELLING AND MODEL ANALYSIS 14 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Problem Statement of Model Parameter Estimation Given:   System model:   Set of measured data:   Loss function: Compute: an estimate such that

PROCESS MODELLING AND MODEL ANALYSIS 15 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Problem Statement of Model Structure Estimation Given:   System model: (not parametrized)   Set of measured data:   Loss function: Compute: an estimate such that + “candidate structures” in

PROCESS MODELLING AND MODEL ANALYSIS 16 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Least Squares (LS) Parameter Estimation Given:   System model: linear in p, single y (M)   Measured data:   Loss function: Compute: an estimate such that

PROCESS MODELLING AND MODEL ANALYSIS 17 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Properties of LS Parameter Estimation Estimation: with Gaussian measurement errors :   unbiased:   covariance matrix:

PROCESS MODELLING AND MODEL ANALYSIS 18 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Assessing the Fit Residuals are independent and   residual tests   correlation coefficient measures

PROCESS MODELLING AND MODEL ANALYSIS 19 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Confidence Regions and Intervals Individual confidence intervals:

PROCESS MODELLING AND MODEL ANALYSIS 20 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences LS Parameter Estimation for Nonlinear Models Solution   Transformation into linear form   Solution by (numerical) optimization   Properties has lost its nice properties - non-normally distributed - confidence region and confidence intervals are not symmetric - unbiased

PROCESS MODELLING AND MODEL ANALYSIS 21 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Confidence Interval for Nonlinear Parameter Estimation Sum-of-squares as a function of a parameter

PROCESS MODELLING AND MODEL ANALYSIS 22 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Static Models Linear in Parameters General form Examples

PROCESS MODELLING AND MODEL ANALYSIS 23 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Identification: Model Parameter and Structure Estimation of Dynamic Models Properties of the estimation problem variables (y and x) are time dependent ordered x : present and past inputs and outputs measurement errors on both y and x Steps 1. sampling continuous time models 2. estimation

PROCESS MODELLING AND MODEL ANALYSIS 24 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Parameter Estimation of Dynamic Models Linear in Parameters General form of the input-output model LS parameter estimation with

PROCESS MODELLING AND MODEL ANALYSIS 25 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Parameter Estimation of Nonlinear Dynamic Models

PROCESS MODELLING AND MODEL ANALYSIS 26 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Statistical Model Validation via Parameter Estimation Conceptual Problem Statement Given:  a calibrated model  validation data (measured data)  measure of fit (loss function) Question: Is the calibrated model “good enough” for the purpose? (Does it reproduce the data well?)

PROCESS MODELLING AND MODEL ANALYSIS 27 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences A Systematic Modelling Procedure Problem definition Controlling factors Problem data Model construction Model solution Model verification Model calibration & validation