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Empirical Modeling. Introduction Regression models First-order transfer function models Second-order transfer function models Integrating models Matlab System Identification Toolbox. Motivation. Fundamental models Derived from conservation principles Typically comprised of ODEs

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empirical modeling
Empirical Modeling
  • Introduction
  • Regression models
  • First-order transfer function models
  • Second-order transfer function models
  • Integrating models
  • Matlab System Identification Toolbox
motivation
Motivation
  • Fundamental models
    • Derived from conservation principles
    • Typically comprised of ODEs
    • Preferred modeling approach when possible
  • Limitations of fundamental modeling
    • Often lack fundamental knowledge of process
    • Unknown parameters must be determined
    • Complex models may not be suitable for controller design
  • Alternative approach
    • Derive model directly from process data
    • Procedure known as process identification
    • Yields empirical models
process identification
Process Identification
  • Basic idea
    • Vary process input u(t)
    • Collect measurements of the process output y(t)
    • Use data to construct dynamic model M relating u(t) and y(t)
    • Goal is to obtain the simplest model possible
  • Limitations
    • Model only represents process dynamics over range of data collected
    • No fundamental knowledge is gained
general modeling procedure
General Modeling Procedure
  • Formulate model objectives
  • Select input and output variables
  • Develop plant testing plan and collect data
  • Analyze dataset and remove “bad” data
  • Select model structure
  • Estimate unknown model parameters by regressing the available data
  • Validate model using data not used for regression
linear regression models
Linear Regression Models
  • Steady-state data: (u1,Y1), (u2,Y2),…, (uN,YN)
  • Assume linear steady-state model structure
  • Least-squares problem
  • Solution:
linearly parameterized models
Linearly Parameterized Models
  • Can apply linear regression if the unknown parameters appear linearly
  • Model structure
  • Least-squares problem
  • Solution:
first order models
First-Order Models
  • Model structure
  • Step response:
  • Gain calculation
  • Time constant calculation
first order model example
First-Order Model Example
  • Gain calculation
  • Time constant calculation
  • Model
first order plus time delay models
First-Order Plus Time Delay Models
  • Model structure
  • Calculation of q and t
    • Determine times when output has reached 35.3% (t35) and 85.3% (t85) of its final value with the time delay removed
    • Calculate q and t
    • The inflection-free method is preferred
second order models
Second-Order Models
  • Model structure
  • Estimate q from step response
  • Calculation of z and t
    • Determine times when output has reached 20% (t20) and 60% (t60) of its final value with the time delay removed
    • Calculate z and t from graph
second order model example
Second-Order Model Example
  • Normalized step response data
    • Assume K is known and q = 0
  • Time constant calculation
integrating models
Integrating Models
  • Model structure
  • Step response
  • Calculation of K
    • Select two times t1 and t2
    • Compute
matlab system identification toolbox
Matlab System Identification Toolbox
  • Data import and processing: represent, process, analyze and manipulate data
  • Linear model identification: estimate transfer function and state-space models from time domain data
  • Analysis: validate and analyze models by comparing model output, computing parameter confidence intervals and prediction errors
  • Simulation and prediction: simulate and predict linear model output
  • Not used in this class