Empirical Modeling

1. Empirical Modeling • Introduction • Regression models • First-order transfer function models • Second-order transfer function models • Integrating models • Matlab System Identification Toolbox

2. 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

3. 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

4. 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

5. Linear Regression Models • Steady-state data: (u1,Y1), (u2,Y2),…, (uN,YN) • Assume linear steady-state model structure • Least-squares problem • Solution:

6. Linearly Parameterized Models • Can apply linear regression if the unknown parameters appear linearly • Model structure • Least-squares problem • Solution:

7. First-Order Models • Model structure • Step response: • Gain calculation • Time constant calculation

8. First-Order Models

9. First-Order Model Example • Gain calculation • Time constant calculation • Model

10. 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

11. First-Order Plus Time Delay Models

12. 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

13. Second-Order Model Example • Normalized step response data • Assume K is known and q = 0 • Time constant calculation

14. Integrating Models • Model structure • Step response • Calculation of K • Select two times t1 and t2 • Compute

15. 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