Process Control: Designing Process and Control Systems for Dynamic Performance

Chapter 4. Modelling and Analysis for Process ControlCopyright Thomas Marlin 2013The copyright holder provides a royalty-free license for use of this material at non-profit educational institutions

CHAPTER 4 : MODELLING & ANALYSIS FOR PROCESS CONTROLWhen I complete this chapter, I want to be able to do the following.Analytically solve linear dynamic models of first and second orderExpress dynamic models as transfer functionsPredict important features of dynamic behavior from model without solving

Outline of the lesson.Laplace transformSolve linear dynamic modelsTransfer function model structureQualitative features directly from modelFrequency responseWorkshopCHAPTER 4 : MODELLING & ANALYSIS FOR PROCESS CONTROL

WHY WE NEED MORE DYNAMIC MODELLINGI can model this; what more do I need?I would like tomodel elements individuallycombine as neededdetermine key dynamic features w/o solving

WHY WE NEED MORE DYNAMIC MODELLINGI would like tomodel elements individuallyThis will be a transfer functionNow, I can combine elements to modelmany process structures

WHY WE NEED MORE DYNAMIC MODELLINGNow, I can combine elements to modelmany process structuresEven more amazing,I can combine toderive a simplifiedmodel!

THE FIRST STEP: LAPLACE TRANSFORMStep Change at t=0: Same as constant for t=0 to t=t=0f(t)=0

THE FIRST STEP: LAPLACE TRANSFORMWe have seenthis term often! Its thestep response toa first orderdynamic system.

THE FIRST STEP: LAPLACE TRANSFORMLets consider plug flow through a pipe. Plug flow has no backmixing; we can think of this a a hockey puck traveling in a pipe.

What is the dynamic response of the outlet fluid property (e.g., concentration) to a step change in the inlet fluid property?Lets learn a newdynamic response& its LaplaceTransform

THE FIRST STEP: LAPLACE TRANSFORMLets learn a newdynamic response& its LaplaceTransformtimeXinXout

THE FIRST STEP: LAPLACE TRANSFORMLets learn a newdynamic response& its LaplaceTransformIs this a dead time?

What is thevalue?

THE FIRST STEP: LAPLACE TRANSFORMLets learn a newdynamic response& its LaplaceTransformThe dynamic model for dead time isThe Laplace transform for a variable after dead time isOur plants have pipes. We willuse this a lot!

THE FIRST STEP: LAPLACE TRANSFORMWe need the Laplace transform of derivatives for solving dynamic models.First derivative:General:

SOLVING MODELS USING THE LAPLACE TRANSFORMTextbook Example 3.1: The CSTR (or mixing tank) experiences a step in feed composition with all other variables are constant. Determine the dynamic response.(Well solve this in class.)I hope we get the same answer as with theintegrating factor!

SOLVING MODELS USING THE LAPLACE TRANSFORMTextbook Example 3.2Two isothermal CSTRs are initially at steady state and experience a step change to the feed composition to the first tank. Formulate the model for CA2. (Well solve this in class.)Much easier thanintegrating factor!

SOLVING MODELS USING THE LAPLACE TRANSFORMTextbook Example 3.5: The feed composition experiences a step. All other variables are constant. Determine the dynamic response of CA. (Well solve this in class.)

TRANSFER FUNCTIONS: MODELS VALID FOR ANY INPUT FUNCTIONLets rearrange the Laplace transform of a dynamic modelY(s)X(s)G(s)Y(s) = G(s) X(s)A TRANSFER FUNCTION is the output variable, Y(s), divided by the input variable, X(s), with all initial conditions zero.

G(s) = Y(s)/X(s)

TRANSFER FUNCTIONS: MODELS VALID FOR ANY INPUT FUNCTIONY(s)X(s)G(s)G(s) = Y(s)/ X(s)How do we achieve zero initial conditions for every model?We dont have primes on the variables; why?Is this restricted to a step input?What about non-linear models?How many inputs and outputs?

TRANSFER FUNCTIONS: MODELS VALID FOR ANY INPUT FUNCTIONY(s)X(s)G(s)G(s) = Y(s)/ X(s)Some examples:

TRANSFER FUNCTIONS: MODELS VALID FOR ANY INPUT FUNCTIONY(s)X(s)G(s)G(s) = Y(s)/ X(s)Why are we doing this?To torture students.We have individual models that we can combine easily - algebraically.We can determine lots of information about the system without solving the dynamic model.I chose thefirst answer!

TRANSFER FUNCTIONS: MODELS VALID FOR ANY INPUT FUNCTIONT(Time in seconds)Lets see how tocombine models

TRANSFER FUNCTIONS: MODELS VALID FOR ANY INPUT FUNCTIONThe BLOCK DIAGRAMGvalve(s)Gtank2(s)Gtank1(s)Gsensor(s)v(s)F0(s)T1(s)T2(s)Tmeas(s)Its a picture of the model equations!Individual models can be replaced easilyHelpful visualizationCause-effect by arrows

TRANSFER FUNCTIONS: MODELS VALID FOR ANY INPUT FUNCTION Combine using BLOCK DIAGRAM ALGEBRAGvalve(s)Gtank2(s)Gtank1(s)Gsensor(s)v(s)F0(s)T1(s)T2(s)Tmeas(s)G(s)v(s)Tmeas(s)

TRANSFER FUNCTIONS: MODELS VALID FOR ANY INPUT FUNCTIONKey rules for BLOCK DIAGRAM ALGEBRA

QUALITATIVE FEATURES W/O SOLVINGFINAL VALUE THEOREM: Evaluate the final valve of the output of a dynamic model without solving for the entire transient response.Example for first order system

What about dynamics can we determinewithout solving?QUALITATIVE FEATURES W/O SOLVINGWe can use partial fraction expansion to prove the following key result.Y(s) = G(s)X(s) = [N(s)/D(s)]X(s) = C1/(s-1) + C2/(s-2) + ...With i the solution to the denominator of the transfer function being zero, D(s) = 0.Real, distinct iReal, repeated iComplex iq is Re(i)

QUALITATIVE FEATURES W/O SOLVINGWith i the solutions to D(s) = 0, which is a polynomial.1.If all i are ???, Y(t) is stableIf any one i is ???, Y(t) is unstable2.If all i are ???, Y(t) is overdamped (does not oscillate)If one pair of i are ???, Y(t) is underdampedComplete statements based on equation.

QUALITATIVE FEATURES W/O SOLVINGWith i the solutions to D(s) = 0, which is a polynomial.1.If all real [i] are < 0, Y(t) is stableIf any one real [i] is 0, Y(t) is unstable2.If all i are real, Y(t) is overdamped (does not oscillate)If one pair of i are complex, Y(t) is underdamped

(Well solve this in class.)QUALITATIVE FEATURES W/O SOLVING1.Is this system stable?2.Is this system over- or underdamped?3.What is the order of the system?(Order = the number of derivatives between the input and output variables)4.What is the steady-state gain?Withoutsolving!

QUALITATIVE FEATURES W/O SOLVINGFREQUENCY RESPONSE:The response to a sine input of the output variable is of great practical importance. Why?Sine inputs almost never occur. However, many periodic disturbances occur and other inputs can be represented by a combination of sines.For a process without control, we want a sine input to have a small effect on the output.

QUALITATIVE FEATURES W/O SOLVINGinputoutputBAPPAmplitude ratio = |Y(t)| max / |X(t)| maxPhase angle = phase difference between input and output

QUALITATIVE FEATURES W/O SOLVINGAmplitude ratio = |Y(t)| max / |X(t)| maxPhase angle = phase difference between input and outputFor linear systems, we can evaluate directly using transfer function! Set s = j, with = frequency and j = complex variable.These calculations are tedious by hand but easily performed in standard programming languages.

QUALITATIVE FEATURES W/O SOLVINGExample 4.15 Frequency response of mixing tank. Time-domain behavior.Bode Plot - Shows frequency response for a range of frequencies

Log (AR) vs log()Phase angle vs log()

QUALITATIVE FEATURES W/O SOLVINGSine disturbance with amplitude = 1 mol/m3frequency = 0.20 rad/minMust have fluctuations < 0.050 mol/m3CA2Using equations for the frequency response amplitude ratioNot acceptable. We needto reduce the variability.How about feedbackcontrol?Data from 2 CSTRs

OVERVIEW OF ANALYSIS METHODSWe can determine individual modelsand combine1. System order2. Final Value3. Stability4. Damping5. Frequency responseWe can determine these features withoutsolving for the entire transient!!Transfer function and block diagram

We can use a standard modelling procedure tofocus ourcreativity!Combining Chapters 3 and 4

Flowchart of Modeling Method

Goal:Assumptions:Data:

Variable(s): related to goals

System: volume within which variables are independent of position

Fundamental Balance: e.g. material, energy

Check D.O.F.

Is model linear?

Expand in Taylor Series

DOF = 0

Another equation: Fundamental balanceConstitutive equations

DOF 0

No

Express in deviation variables

Group parameters to evaluate [gains (K), time-constants (), dead-times()]

Take Laplace transform

Substitute specific input, e.g., step, and solve for output

Analytical solution(step)

Numerical solution

Analyze the model for:- causality order stability damping

Yes

Combine several models into integrated system

Too small to read here - check it out in the textbook!

CHAPTER 4: MODELLING & ANALYSIS WORKSHOP 1Example 3.6 The tank with a drain has a continuous flow in and out. It has achieved initial steady state when a step decrease occurs to the flow in. Determine the level as a function of time.Solve the linearized model using Laplace transforms

CHAPTER 4: MODELLING & ANALYSIS WORKSHOP 21. System order2. Final Value3. Stability4. Damping5. Frequency responseThe dynamic model for a non-isothermal CSTR is derived in Appendix C. A specific example has the following transfer function.Determine the features in the table for this system.

CHAPTER 4: MODELLING & ANALYSIS WORKSHOP 3Answer the following using the MATLAB program S_LOOP.Using the transfer function derived in Example 4.9, determine the frequency response for CA0 CA2. Check one point on the plot by hand calculation.

CHAPTER 4: MODELLING & ANALYSIS WORKSHOP 4We often measure pressure for process monitoring and control. Explain three principles for pressure sensors, select one for P1 and explain your choice.

CHAPTER 4 : MODELLING & ANALYSIS FOR PROCESS CONTROLWhen I complete this chapter, I want to be able to do the following.Analytically solve linear dynamic models of first and second orderExpress dynamic models as transfer functionsPredict important features of dynamic behavior from model without solvingLots of improvement, but we need some more study!Read the textbookReview the notes, especially learning goals and workshopTry out the self-study suggestionsNaturally, well have an assignment!

LEARNING RESOURCESSITE PC-EDUCATION WEB - Instrumentation Notes- Interactive Learning Module (Chapter 4)

- Tutorials (Chapter 4)Software Laboratory- S_LOOP programwww.pc-education.mcmaster.ca/

SUGGESTIONS FOR SELF-STUDY1.Why are variables expressed as deviation variables when we develop transfer functions?2.Discuss the difference between a second order reaction and a second order dynamic model.3.For a sine input to a process, is the output a sine for a a. Linear plant?b. Non-linear plant?4. Is the amplitude ratio of a plant always equal to or greater than the steady-state gain?

SUGGESTIONS FOR SELF-STUDY5.Calculate the frequency response for the model in Workshop 2 using S_LOOP. Discuss the results.6.Decide whether a linearized model should be used for the fired heater fora.A 3% increase in the fuel flow rate.b.A 2% change in the feed flow rate.c.Start up from ambient temperature.d.Emergency stoppage of fuel flow to 0.0.fuelfeedair