Parameter Estimation

1
Parameter Estimation

• Basic principles
• Parameter estimation with steady-state data
• Exothermic chemical reactor
• Parameter estimation with dynamic data

2
Motivation

• Mathematical models
• Derived from conservation principles
• Involve unknown parameters (q)
• Reaction rate constants, activation energies,
  heat transfer coefficients, relative
  volatilities, diffusion coefficients, etc.
• Often unavailable from open literature
• Parameter estimation
• Laboratory experiments expensive time consuming
• Alternative is to estimate unknown parameters
  from available data
• Generate data directly from process being modeled

3
Parameter Estimation Problems

• Optimization formulation
• Objective function f(y,u,q)
• Least-squares error between experimental data
  model predictions
• Unknown parameters q
• Decision variables
• Input variables u
• Adjusted to generate informative data sets for
  parameter estimation
• Equality constraints h(y,u,q)
• Algebraic model equations
• ODE models must be discretized in time to obtain
  algebraic equations
• Inequality constraints g(y,u,q)
• Usually bounds on unknown parameters

4
Steady-State Parameter Estimation

• Steady-state model
• Experiments
• Perform N experiments by varying inputs (u)
• More experiments than the number of unknown
  parameters N gt p
• Collect steady-state data for each experiment (m
  measured)
• Least-squares objective function
• Prediction error for jth experiment
• Sum of squared errors
• Diagonal matrix Q used to adjust weighting
  according to variable scaling measurement
  confidence

5
Linearly Parameterized Models

• Model equations
• Parameter estimation problem
• Objective function quadratic function of
  parameters
• Constraints linear functions of parameters
• Can be formulated as quadratic program

6
Least-Square Solution

• Prediction error
• Variable definitions
• Overdetermined least-squares problem

7
Nonlinearly Parameterized Models

• Model equations
• Parameter estimation problem
• Nonlinear objective function
• Nonlinear equality constraints
• Requires solution of nonlinear program
• More difficult problem than for linearly
  parameterized models

8
Exothermic CSTR

• Unknown model parameters
• Activation energy (E), frequency factor (k0)
  heat transfer coefficient (U)
• Subject to lower upper bounds
• Data collection
• Select N different residence time tj V/qj
• Collect steady-state concentration and
  temperature data
• Steady-state model equations
• Nonlinear programming problem

9
Dynamic Parameter Estimation

• Dynamic data provides more information than
  steady-state data
• Dynamic model
• Experiments
• Perform N experiments by varying inputs (u)
• Collect dynamic data at M time points for each
  experiment
• Least-squares objective function
• Prediction error
• Sum of squared errors

10
Dynamic Model Discretization

• Motivation
• Constrained optimization codes require algebraic
  model equations
• Nonlinear ODE models can rarely be solved
  analytically
• Discretize time derivatives to obtain algebraic
  equations
• Simple example
• Each ODE converted into M algebraic equations
• Each y(ti) is considered as a variable
• More accurate stable discretizations preferred
• Parameter estimation
• Consider discretized algebraic equations as
  equality constraints
• Generates large nonlinear optimization problems