CHAPTER 6: EMPIRICAL MODELLING We have invested a lot of effort to - PowerPoint PPT Presentation

CHAPTER 6: EMPIRICAL MODELLING We have invested a lot of effort to learn fundamental modelling. Why are we now learning about an empirical approach? TRUE/FALSE QUESTIONS • We have all data needed to develop a fundamental model of a complex process • We have the time to develop a fundamental model of a complex process • Experiments are easy to perform in a chemical process • We need very accurate models for control engineering

EMPIRICAL MODEL BUILDING PROCEDURE Start A priori knowledge Experimental Design Not just process control Plant Experimentation Determine Model Structure Parameter Estimation Diagnostic Evaluation Alternative Model Verification data Complete

EMPIRICAL MODEL BUILDING PROCEDURE Process reaction curve - Method I 45 15 S = maximum slope 35 11 input variable in deviation (% open) output variable in deviation (K) ∆ 25 7 15 3 θ 5 -1 δ -5 -5 0 10 20 30 40 time (min) Data is plotted in deviation variables

EMPIRICAL MODEL BUILDING PROCEDURE Process reaction curve - Method II 45 15 35 11 input variable in deviation (% open) output variable in deviation (K) 0.63 ∆ ∆ 25 7 0.28 ∆ 15 3 5 -1 δ t 28% t 63% -5 -5 0 10 20 30 40 time (min) Data is plotted in deviation variables

55 Let’s get get out the calculator and practice with this experimental data. 51 output variable, degrees C input variable, % open 47 43 55 39 45 0 10 20 30 40 time

EMPIRICAL MODEL BUILDING PROCEDURE Process reaction curve Start 45 15 Experimental Design 35 11 input variable in deviation (% open) output variable in deviation (K) Plant Experimentation 25 7 15 3 Determine Model Structure 5 -1 Parameter Estimation -5 -5 0 10 20 30 40 time (min) Diagnostic Evaluation Model Verification Is this a well designed experiment? Complete

EMPIRICAL MODEL BUILDING PROCEDURE Process reaction curve Start 45 15 Experimental Design 35 11 input variable in deviation (% open) output variable in deviation (K) Plant Experimentation 25 7 15 3 Determine Model Structure 5 -1 Parameter Estimation -5 -5 0 10 20 30 40 time (min) Diagnostic Evaluation Input should be close to a perfect Model Verification step; this was basis of equations. If not, cannot use data for process Complete reaction curve.

EMPIRICAL MODEL BUILDING PROCEDURE Process reaction curve 45 15 Start 35 11 Experimental Design output variable, degrees C input variable, % open 25 7 Plant Experimentation 15 3 Determine Model Structure 5 -1 Parameter Estimation -5 -5 Diagnostic Evaluation 0 10 20 30 40 time Model Verification Should we use this data? Complete

EMPIRICAL MODEL BUILDING PROCEDURE Process reaction curve 45 15 Start 35 11 output variable, degrees C Experimental Design input variable, % open 25 7 Plant Experimentation 15 3 Determine Model Structure 5 -1 -5 -5 Parameter Estimation 0 10 20 30 40 time Diagnostic Evaluation The output must be “moved” Model Verification enough. Rule of thumb: Signal/noise > 5 Complete

EMPIRICAL MODEL BUILDING PROCEDURE Process reaction curve Start 45 10 Experimental Design 35 6 output variable, degrees C Plant Experimentation input variable, % open 25 2 Determine Model Structure 15 -2 Parameter Estimation 5 -6 Diagnostic Evaluation -5 -10 0 20 40 60 80 Model Verification time Should we use this data? Complete

EMPIRICAL MODEL BUILDING PROCEDURE Output did not Process reaction curve return close to the Start initial value, 45 10 although input Experimental Design returned to initial 35 6 value output variable, degrees C Plant Experimentation input variable, % open 25 2 Determine Model Structure 15 -2 Parameter Estimation 5 -6 Diagnostic Evaluation -5 -10 0 20 40 60 80 Model Verification time This is a good experimental design; it checks Complete for disturbances

EMPIRICAL MODEL BUILDING PROCEDURE Process reaction curve Start Plot measured vs predicted Experimental Design 45 15 Plant Experimentation measured 35 11 Determine Model Structure output variable, degrees C input variable, % open 25 7 Parameter Estimation 15 3 Diagnostic Evaluation predicted 5 -1 Model Verification -5 -5 Complete 0 10 20 30 40 time

EMPIRICAL MODEL BUILDING PROCEDURE Statistical method Provides much more general approach that is not restricted to • step input • first order with dead time model • single experiment • “large” perturbation • attaining steady-state at end of experiment Requires • more complex calculations

EMPIRICAL MODEL BUILDING PROCEDURE Statistical method • The basic idea is to formulate the model so that regression can be used to evaluate the parameters. • We will do this for a first order plus dead time model, although the method is much more general. • How do we do this for the model below? − θ s K p e Y ( s ) dY ( t ) = τ + = − θ Y ( t ) K X ( t ) p τ + X ( s ) s 1 dt

EMPIRICAL MODEL BUILDING PROCEDURE Statistical method We have discrete measurements, so let’s express the model as a difference equation, with the next prediction based on current and past measurements. ( ) ( ) ( ) measured ' = ' + ' Y a Y b X + − Γ i 1 i i predicted measured − ∆ τ t / = a e − ∆ τ = − t / b K ( 1 e ) p Γ = θ ∆ / t

EMPIRICAL MODEL BUILDING PROCEDURE [ ] ( ) ( ) 2 ' − ' ∑ min Y Y i i predicted measured i 45 15 Now, we can solve a standard 35 11 regression problem output variable, degrees C input variable, % open to minimize the 25 7 sum of squares of deviation between 15 3 prediction and measurements. 5 -1 Details are in the book. -5 -5 0 10 20 30 40 time

EMPIRICAL MODEL BUILDING PROCEDURE Match the method to the application. Feature Process reaction curve Statistical method Input magnitude Signal/noise > 5 Can be much smaller Experiment duration Reach steady state Steady state not required Input change shape Nearly perfect step Arbitrary, sufficient “information” required Model structure First order with dead time General linear dynamic model Accuracy with Poor with significant disturbance Poor with significant disturbance unmeasured disturbances Diagnostics Plot prediction vs data Plot residuals Calculations simple Requires spreadsheet or other computer program

EMPIRICAL MODEL BUILDING How accurate are empirical models? • Linear approximations of non-linear processes • Noise and unmeasured disturbances influence data • Lack of consistency in graphical method • lack of perfect implementation of valve change • sensor errors Let’s say that each parameter has an error ± 20%. Is that good enough for future applications?