7.0 context and direction7.1 big and slow - high-order overdamped systems7.2 the FODT approximation to high-order step response7.3 dead time is delay7.4 dead time and lag are different7.5 frequency response of a dead time process7.6 the FODT model7.7 Bode plot of FODT7.8 identification - obtaining an FODT to represent a process 7.9 digital calculation of dead time7.10 an example process with dead time7.11 the control scheme7.12 sensor transmitter range7.13 valve saturation7.14 proportional gain and proportional band7.15 integral time and reset rate7.16 reset windup in integral mode7.17 protecting set point changes from derivative spikes7.18 filtering the derivative mode7.19 closed loop transfer function7.20 Bode criterion for closed loop stability7.21 tuning the controller via correlationsZeigler-Nichols7.22 conclusion7.23 referencesSpring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 7: High Order Overdamped Processes 7.0 context and direction Chemical processing plants are characterized by large time constants and time delays. For control engineering, we can often approximate these high-order systems by the FODT (first-order-dead-time) model. Dead time in a process increases the difficulty of controlling it. DYNAMIC SYSTEM BEHAVIOR 7.1 big and slow - high-order overdamped systems We began our study of process control by considering a mixed tank. Applying a material or energy balance to a well-mixed tank produces a first-order lag system. We subsequently combined two balances to produce a second-order system. In one case, two material balances described storage of material in two tanks. In another case, a single tank stored both material and energy. Energy and material balances show that the tank causes a dynamic lag between input and output, because it takes time to adjust the amount of mass or energy distributed throughout the tank. We might thus expect that more storage elements would lead to higher-order behavior, and require higher-order equations to describe them. The classic illustration of a high-order system is a set of n tanks in series: each tank feeds the next, and a change in the inlet stream composition CA0 must propagate through multiple tanks to be felt at the output CAn. The individual tank models are 'Ai'1Ai'AiiFCFCCVdtd−=− (7.1-1) They are combined by eliminating the interior stream variables to produce a single transfer function between input and output. ()()()1s1s1s1)s(C)s(Cn21'0A'An+τ+τ+τ=" (7.1-2) Let us illustrate high-order behavior and (7.1-2) by first imagining a single well-mixed overflow tank of time constant τ. If we introduce a step increase in the inlet concentration, we will (by the well-mixed assumption) immediately detect a rise in the outlet stream – the familiar first-order lag response. If we have instead two tanks in series, each half the volume of the original, we will detect a second-order, sigmoid response at the outlet. revised 2006 Mar 29 1Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 7: High Order Overdamped Processes Each tank has a smaller individual time constant, and their sum is the time constant τ of the original tank. If we continue to increase the number of tanks in the series, always maintaining the total volume, we observe a slower initial response with a faster rise around the time constant. This behavior is shown in Figure 7.1-1. 00.20.40.60.811.20 0.5 1 1.5 2 2.5 3 3.5 4 4.5time/taustep response1261200.20.40.60.811.20 0.5 1 1.5 2 2.5 3 3.5 4 4.5time/taustep response12612 Figure 7.1-1. Step response for tanks in series; equal time constants revised 2006 Mar 29 2Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 7: High Order Overdamped Processes The step response shows that high-order systems have a longer start-up period before rising toward the final value. 7.2 the FODT approximation to high-order step response There may be occasions when detailed dynamic analysis of a process is warranted. Often, however, it is sufficient to obtain a simplified dynamic model that gives a reasonable approximation to the process behavior. Figure 7.1-1 indicates that high-order step responses might be represented as a first-order rise following a period of delay. The dynamic model that would behave this way is called the FODT (first-order-dead-time) model; it turns out to be suitable for describing the dynamic response of many chemical processes. 7.3 dead time is delay Before we examine the FODT model, we will look at dead time by itself. Chemical processes require that material be moved from one location to another: in conduits, on conveyer belts, through vessels. The transportation time is finite; it implies a delay between the onset of a disturbance at one location and its observation at another. This delay is often called dead time; it is familiar to anyone who has waited at the faucet for the hot water to arrive. Consider a pipe carrying a liquid. A pulse of solute added at the entrance will be observed at the exit only after the solute is transported through the pipe. tdtd+ θtimesignalx(t) seen by observer at entrance of pipe y(t) seen by observer at exit of pipetdtd+ θtimesignalx(t) seen by observer at entrance of pipe y(t) seen by observer at exit of pipe The transit time depends on the liquid velocity and the length of the pipe. The figure indicates faithful transmission of the input signal x(t) from inlet to outlet, as if every particle in the pipe moved at the same velocity. However, in real chemical processes the solute pulse y(t) would become distorted through diffusive and dispersive effects. Nonetheless, a simple description of transport using the average fluid velocity is often sufficient to represent dead time in a process: revised 2006 Mar 29 3Spring 2006 Process Dynamics, Operations, and Control 10.450 Lesson 7: High Order Overdamped Processes FVvL==θ (7.3-1) Thus the dead time is the residence time in the pipe. Consider again Figure 7.1-1 and the series of tanks. If taken to the limit of an infinite number of tanks, each infinitesimally small, we finally obtain a pure delay, in which at time τ the full step disturbance appears at the outlet. 7.4 dead time and lag are different In casual conversation, one might not distinguish between lag and delay; however, in process control these two terms have distinct meanings. A lag process is illustrated by a mixed