2Common simple dynamic systems - First order -Second order Outline of the lesson.Common simple dynamic systems- First order -Second order- Dead time - (Non) Self-regulatory
3When I complete this chapter, I want to be able to do the following. Predict output for typical inputs for common dynamic systems
43.1. First-Order Differential Equation Models The model can be rearranged aswhere τ is the time constant and K is the steady-state gain.The Laplace transform is
53.1.1. Step Response of a First-Order Model In the previous lecture, we have learned about the step response of first order systems.
63.1.2. Impulse Response of a First-Order Model Consider an impulse input, u(t) = δ (t), and U(s) = 1; the output is nowThe time-domain solution iswhich implies that the output rises instantaneously to some value at t = 0 and then decays exponentially to zero.
7Integrating ProcessWhen the coefﬁcient a0 = 0 in the first order differential equation , we getwhere K = (b / a1). Here, the pole of the transfer function G(s) is at the origin, s = 0.
8Integrating ProcessThe solution of the Equation , could be written immediately without any transform asThis 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.
9SIMPLE PROCESS SYSTEMS: INTEGRATOR pumpvalveLevel sensorLiquid-filledtankPlants 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.
10SIMPLE PROCESS SYSTEMS: INTEGRATOR Plot the level for this scenario pumpvalveLevel sensorLiquid-filledtankPlot the level for this scenarioFinFouttime
11SIMPLE PROCESS SYSTEMS: INTEGRATOR pumpvalveLevel sensorLiquid-filledtankLevelFinFouttime
12SIMPLE PROCESS SYSTEMS: INTEGRATOR pumpvalveLevel sensorLiquid-filledtankLet’s look aheadto when weapply control.Non-self-regulatory variables tend to “drift” far from desired values.We must control these variables.
133.2. Second-Order Differential Equation Models We have not encountered examples with a second-order equation, especially one that exhibits oscillatory behavior.One reason is that processing equipment tends to be self-regulating.An oscillatory behavior is most often the result of implementing a controller.For now, this section provides several important deﬁnitions.
143.2. Second-Order Transfer Function Models where ωn is the natural (undamped) frequency, ζ is the damping ratio or coefﬁcient, K is the steady-state gain, and τ is the natural period of oscillation, where τ = 1/ωn.The characteristic equation isWhich provides the poles
15SIMPLE PROCESS SYSTEMS: 2nd ORDER 10203040506070800.20.40.60.81TimeControlled VariableManipulated Variable1001201401601802000.51.5overdampedunderdamped
16WORKSHOPFour systems experienced an impulse input at t=2. Explain what you can learn about each system (dynamic model) from the figures below.51015202530123output(a)-1(b)time(c)0.51.52.5(d)
173.4 Processes with dead time Many chemical processes involve a time delay between the input and the 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.In controller design, the measured output will not contain the most current information, and hence systems with dead time can be difficult to control.
18Let’s learn a newdynamic response& its LaplaceTransformLet’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?
19THE FIRST STEP: LAPLACE TRANSFORM = dead timeWhat is the value ofdead time forplug flow?XoutXintime
20THE FIRST STEP: LAPLACE TRANSFORM Is this adead time?What is thevalue?12345678910-0.50.5timeY, outlet from dead timeX, inlet to dead time
21THE FIRST STEP: LAPLACE TRANSFORM Our plants havepipes. We willuse this a lot!The dynamic model for dead time isThe Laplace transform for a variable after dead time is
22Pade approximation of the time delay There several methods to approximate the dead time as a ratio of two polynomials in s.On such method is the first-order Pade approximation.
23Example 3.2Use the ﬁrst-order Pade approximation to plot the unit-step response of the ﬁrst order with a dead-time function: Making use of the dirst order Pade approximation, we can construct a plot with the approximation
24Matlab codeth = 3; P1 = tf([-th/2 1],[th/2 1]); % First-order Padé approximation t = 0:0.5:50; taup = 10; G1 = tf(1,[taup 1]); y1 = step(G1*P1,t); % y1 is first order with Padé approximation of % dead time y2 = step(G1,t); t2 = t+th; % Shift the time axis for the actual time-delay function plot(t,y1,t2,y2,’r’);
25The approximation is very good except near t = 0, where the approximate response dips below. This behavior has to do with the ﬁrst-order Pade approximation, and we can improve the result with a second-order Pade approximation.