1. Lecture 3: Common Dynamic Systems 1

2. Identify common dynamic elements:  First order  Integrating (Non self-regulatory)  Second order  Dead time Predict the response for typical inputs for these systems. 2 Objectives

3. First-order models The first-order model: where τ is the time constant and K is the steady-state gain. The Laplace transform is Previously, we have learned about the step response of first order systems. The first order system tends to be self-regulating as its step response reaches a steady state. 3

4. Impulse response of first-order model Consider an impulse input, u(t) = δ (t), and U(s) = 1; the output is now The time-domain solution is which implies that the output rises instantaneously to some value at t = 0 and then decays exponentially to zero. 4

5. Integrating Process The integrating process can be described by Here, the pole of the transfer function G(s) is at the origin, s = 0. 5

6. Integrating Process The solution of the Equation could be written immediately without any transform as This is called an integrating (also capacitive or non-self- regulating) process. We can associate the name with charging a capacitor or filling up a tank. 6

7. Integrator pumpvalve Level sensor Liquid-filled tank Plants have many inventories whose flows in and out do not depend on the inventory (when we apply no control or manual correction). These systems are often termed “pure integrators” because they integrate the difference between in and out flows. 7

8. Integrator pumpvalve Level sensor Liquid-filled tank F out F in Plot the level for this scenario time 8

9. Integrator pumpvalve Level sensor Liquid-filled tank F out F in time Level 9

10. Integrating processes Non-self-regulatory: output variable tend to “drift” far from desired values. Feedback control is necessary for these processes. 10

11. Second-order models where ω n is the natural (undamped) frequency, ζ is the damping ratio, and K is the steady-state gain. The characteristic equation is Which provides the poles 11

12. Second-order models Processing equipment tends to be self-regulating (this can be a first-order or second-order over-damped process). A second-order process that exhibits oscillatory behavior (under-damped) is most often the result of implementing a controller. 12

13. 2 nd Order Process 13

14. WORKSHOP Four systems experienced an impulse input at t=2. Explain what you can learn about each system (dynamic model) from the figures below. 14

15. Processes with dead time Many chemical processes involve a time delay between input and output. This delay may be due to the time required for a slow chemical sensor to respond or for a fluid to travel down a pipe. A time delay is also called dead time or transport lag. The effect of time delay on feedback control is that the measured output will not contain the most current information, and hence systems with dead time can be difficult to control. 15

16. Let’s 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? Let’s learn a new dynamic response & its Laplace Transform 16

17. The first step: Laplace Transform time U in Y out  = dead time  17

18. The first step: Laplace Transform Is this a dead time? What is the value? 18

19. The first step: Laplace Transform The dynamic model for dead time is The Laplace transform for a variable after dead time is Our plants have pipes. We will use this a lot! 19

20. There are several methods to approximate the dead time as a ratio of two polynomials in s. On such method is the first-order Pade approximation. Pade approximation of the time delay 20

21. Example Use the first-order Pade approximation to plot the unit-step response of the first order with a dead-time transfer function: Making use of the first order Pade approximation, we can construct a plot with the approximation 21

22. The approximation is very good except near t = 0, where the approximate response dips below. A better approximation can be obtained with, e.g., a second-order Pade approximation. 22