Process Control and Optimization

1. Advance Process Control ICarlos VelázquezPharmaceutical Engineering Research LaboratoryERC on Structured Organic Particulate SystemsDepartment of Chemical EngineeringUniversity of Puerto Rico at Mayaguez

2. Process Control and Optimization • Control and optimization are terms that are many times erroneously interchanged. • Control has to do with adjusting flow rates to maintain the controlled variables of the process at specified setpoints. • Optimization chooses the values for key setpoints such that the process operates at the “best” economic conditions.

3. Review of Basic Control

4. Conservation Equations:Mass, Moles, or Energy Balances

5. Laplace Transforms and Transfer Functions • Provide valuable insight into process dynamics and the dynamics of feedback systems. • Provide a major portion of the terminology of the process control profession. • Are NOT generally directly used in the practice of process control.

6. Method for Solving Linear ODE’s using Laplace Transforms

7. Transfer Functions • Defined as G(s) = Y(s)/U(s) • Represents a normalized model of a process, i.e., can be used with any input. • Y(s) and U(s) are both written in deviation variable form. • The form of the transfer function indicates the dynamic behavior of the process.

8. Example of Derivation of a Transfer Function • Dynamic model of CST thermal mixer • Apply equation to the steady state • Substitute deviation variables • Equation in terms of deviation variables.

9. Derivation of a Transfer Function • 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.

10. What if the Process Model is Nonlinear • Before transforming to the deviation variables, linearize the nonlinear equation. • Transform to the deviation variables. • Apply Laplace transform to each term in the equation. • Collect terms and form the desired transfer functions.

11. Use Taylor Series Expansion to Linearize a Nonlinear Equation • This expression provides a linear approximation of y(x) about x=x0. • The closer x is to x0, the more accurate this equation will be. • The more nonlinear that the original equation is, the less accurate this approximation will be.

12. General form of a Transfer Function

13. Dynamic Modeling Approach for Process Control Systems

14. Dynamic Model for Sensors • These equations assume that the sensors behave as a first order process. • The dynamic behavior of the sensor is described by the time constant since the gain is unity • T and L are the actual temperature and level.

15. Dynamic Model for Sensors • The units of KT depend on the input to the sensor and the type of signal of the sensor. • For instance, if input is pressure and the signal from the sensor is in mA, then KT = psig/mA • input : concentration, level, pressure, temperature, force, velocity • Signals : mA, mV, %TO

16. Actuator System • Control Valve • Valve body • Valve actuator • I/P converter • Instrument air system

17. Dynamic Model for Actuators • These equations assume that the actuator behaves as a first order process. • The dynamic behavior of the actuator is described by the time constant since the gain is unity

18. Dynamic Response of an Actuator (First Order Process)

19. Parameter Estimation

21. Estimation of Transfer Functions • Factors involved • Sampling time • Signal-to-noise ratio • Input type

22. 1) Sampling time (T) • If proper sampling is not used, the data could be corrupted by too much noise (fast sampling) or could lack of enough dynamic data (too slow sampling) tp Kp Figure 3.6 Effect of a double pulse and T= 10.0 in the SSE surface of a FO model

23. tp Kp Figure 3.7Effect of a double pulse and T= 100.0 in the SSE surface of a FO model

24. 2) Signal-to-Noise ratio • Measures how big is the signal of the measured variable compared to the signal of the noise

25. tp Kp Figure 3.14Effect of a double pulse and a S/N = 10:1 in the SSE surface of a FO model, T = 1.0

26. tp Kp Figure 3.15 Effect of a double pulse and a S/N = 1:1 in the SSE surface of a FO model, T =1.0

27. 3) Input wave form • Virtually any type of input can be used. The amplitude of the signal is the key factor for an input to be proper.

28. tp Kp Figure 3.1 Effect of a step change in the SSE surface of a FO model

29. tp Kp Figure 3.2 Effect of a pulse in the SSE surface of a FO model

30. tp Kp Figure 3.3 Effect of a double pulse in the SSE surface of a FO model

31. tp Kp Figure 3.4 Effect of sinusoid in the SSE surface of a FO model

32. tp Kp Figure 3.5 Effect of PRBS in the SSE surface of a FO model

33. Feedback Controllers Position Form of the PID Algorithm • Direct acting • Reverse acting

34. Definition of Terms • e(t)- the error from setpoint [e(t)=ysp-ys]. • Kc- the controller gain is a tuning parameter and largely determines the controller aggressiveness. • tI- the reset time is a tuning parameter and determines the amount of integral action. • tD- the derivative time is a tuning parameter and determines the amount of derivative action.

35. Properties of Proportional Action • Closed loop transfer function base on P-only control applied to a first order process. • Properties of P control • Does not change order of process • Closed loop time constant is smaller than open loop tp • Does not eliminate offset.

36. Offset Resulting from P-only Control

37. Properties of Integral Action • Based on first order process • Properties of I control • Offset is eliminated • Increases the order by 1 • As integral action is increased, the process becomes faster, but at the expense of more sustained oscillations

38. Integral Action for the Response of a PI Controller

39. Controller Tuning • Ziegler-Nichols • Integral Criteria • Internal Model Control • Frequency Techniques

41. Stability • Substitution • Root Locus • Frequency techniques

42. Stability of the Control Loop • For a feedback control loop to be stable, all the roots of its characteristic equation must be either negative real numbers or complex numbers with negative real parts.

43. Stability of a Controlled System • Method 1: Direct Substitution • Select a P-Only controller. • Write the characteristic equation in a polynomial form of s. • Substitute s=wui and Kc=Kcu • Solve for Kcu and wu

44. Example Exchanger Sensor/Transmitter Valve P-Only controller

45. Solution Characteristic equation Substituting corresponding TFs Rearranging Substituting s=iwu, and Kc=Kcu

46. Solution The real and imaginary parts have to be each one equal to zero to make the equation equal to zero.

47. Solution • To determine the correct range, first we need to determine the sign of the multiplication of the gain of the process, sensor, and actuator. • If KpKTKV is (+) then Kc must be +. • If KpKTKV is (-) then Kc must be -. • In this case, Kc must be positive, hence Kc = 23.8 is the only physical sound solution

48. Feedback advance strategies

49. Cascade • Main purpose: Reject disturbances that affect an intermediate controlled variable before it hit the main controlled variable. • Results: Improve performance in rejecting some process disturbances.

50. Cascade control scheme