CHE 185 – PROCESS CONTROL AND DYNAMICS

1. CHE 185 – PROCESS CONTROL AND DYNAMICS TRANSFER FUNCTIONS

2. TRANSFER FUNCTIONS • RELATE INPUT (INDEPENDENT VARIABLE) TO OUTPUT (DEPENDENT VARIABLE). • DEFINITION OF THE TRANSFER FUNCTION, G(s), IS THE LaPLACE TRANSFORM OF THE OUTPUT VARIABLE, Y(s), DIVIDED BY THE LaPLACE TRANSFORM OF THE INPUT VARIABLE, U(s), WITH ALL INITIAL CONDITIONS SET TO ZERO. G • TRANSFER FUNCTIONS ONLY APPLY TO LINEAR MODELS

3. TRANSFER FUNCTIONS • A process transfer function, GP(s), is an equation in the Laplace domain that describes the dynamic response of the measured process variable to changes in the manipulated process variable (controller output signal). • the denominator of a transfer function is the characteristic equation of the time domain complementary solution. • roots of the characteristic equation indicate a system’s stability and natural tendency to oscillate.

4. TRANSFER FUNCTIONS • unstable system results if any root has a positive real part, e+ p1tand that term will grow without bound as time tgrows to infinity. • stable system results if all roots have • negative real parts, e- p1t, as these terms all die out (go to zero) as tgrows to infinity. • tendency to oscillate is a consequence of sine and cosine terms in the solution that results from imaginary roots.

5. Transfer function poles • the denominator of a transfer function is the characteristic equation of the time domain complementary solution, and the roots of the denominator of a transfer function are called poles • Poles indicate system stability and tendency to oscillate. • This knowledge will prove useful in control system analysis and design studies

6. transfer function example • FOPDT (first order plus dead time) • Laplacian • Apply initial condition • Arrange into transfer function • Fopdt Complementary equation • Characteristic equation of the time domain complementary solution

7. U(s) Y(s) G(s) Input Output Transfer function VARIABLES IN TRANSFER FUNCTIONS • EXPRESSED AS DEVIATIONS FROM THE INITIAL CONDITIONS • ALL DERIVATIVES • ARE ZERO IF THE INITIAL CONDITIONS ARE AT STEADY STATE • Are DEFINED AS DIFFERENCE BETWEEN VARIABLE AND STEADY STATE VALUE .

8. VARIABLES IN TRANSFER FUNCTIONS • In terms of variation variables: • EXAMPLES 5.1 AND 5.2 SHOW CONVERSION OF GENERAL FORMS OF ODE’s INTO DEVIATION VARIABLE FORM. • Tank level • Cstr composition • EXAMPLES 5.3 and 5.4 SHOW DERIVATION OF TRANSFER FUNCTIONS. • Generic First order equation • Pid controller

9. TRANSFER FUNCTION PROPERTIES • Additive property • Y(s) = G1(s)U1(s)+ G2(s)U2(s) • Multiplicative property • Y2(s) = G1(s)G2(s)U(s) • ODE equivalence

10. COMMON PROCESS INPUTS • Step input • Ramp input • Rectangular pulse input

11. COMMON PROCESS INPUTS • IMPULSE INPUT • Sinusoidal input

12. TRANSFER FUNCTION example • CONCENTRATION CHANGE IN A MIXING TANK AFTER A STEP CHANGE in feed concentration • BASES: WELL-MIXED, CONSTANT FLOW IN, DENSITY OF A SAME AS SOLVENT, AND SYSTEM IS INITIALLY AT STEADY STATE, At t = 0, Ca0 = CA = CAi

13. TRANSFER FUNCTION example • DEFINING EQUATIONS: • Constant volume : • Mass balance for a: • Degrees of freedom • Variables are Ca and F1, and equations are as shown • so dof = 2 – 2 = 0 and system is exactly specified.

14. TRANSFER FUNCTION example • Mathematical solution • Mass balance equation is linear, first order ode that is not separable: • Standard form is: • Time constant :

15. TRANSFER FUNCTION example • Equation can be converted into a separable form by multiplying both sides by the integrating factor:

16. TRANSFER FUNCTION example • Setting up the integrations and solving: where I is an integration constant, or =+I

17. TRANSFER FUNCTION example • The integration constant can be evaluated using the initial condition : • So for a step change: • - • And for this system: • Steady state gain:

18. TRANSFER FUNCTION example • laplace solution • Model in deviation variables: • LaPlace transform for each term: • Initial deviation for tank concentration = 0 • Inlet concentration deviation is constant for step function for t>0:

19. TRANSFER FUNCTION example • Substitution into deviation equation: • The inverse transform of this expression is: ) • And the transfer function is:

20. TRANSFER FUNCTION example cst thermal mixer • Dynamic model of CST thermal mixer – ex. 3.1 • Apply deviation variables • Equation in terms of deviation variables.

21. TRANSFER FUNCTION example cst thermal mixer • Apply Laplace transform to each term considering that only inlet and outlet temperatures change. • Determine the transfer function for the effect of inlet temperature changes on the outlet temperature. • Note that the response is first order.

22. TRANSFER FUNCTIONS • MULTIPLE TRANSFER FUNCTIONS ARE DEVELOPED FOR MULTIPLE INPUTS OR OUTPUTS • LINEARITY ALLOWS THE EFFECTS OF MULTIPLE INPUTS TO BE SUMMED. • ORDER OF TRANSFER FUNCTIONS • IS BASED ON THE HIGHEST DERIVATIVE OF THE OUTPUT VARIABLE IN THE DIFFERENTIAL EQUATION. • THIS IS ALSO REPRESENTED BY THE HIGHEST POWER OF s IN THE DENOMINATOR OF THE TRANSFER FUNCTION.

23. System Order • General transfer function • System order • Order of the denominator polynomial D(s) • Generally equal to the number of ODEs from which G(s) was derived

24. System Order • First-order system • K = steady state gain • τ= time constant • Second-order system • τn= natural period • ζ = damping factor

25. First order systems • Standard equation form • (6.3.2) • So for example 6.2 • becomes • So process gain is 8 and the time constant is 0.5.

26. second order systems • Standard equation form • (6.4.2) • So for example 6.6 becomes • So time constant, τn equals 2 time units, process gain is 2 and damping factor,  = 0.75.

27. High order processes • Combine first order processes in series • Overall model is product of the first order processes • For equal time constants in each step • Example can be trays in a column

28. deadtime • Time Delay in process between change and measurement • Defined as θ • Deadtime transfer function for pure time delay is : • Impact for feedback control can be significant