First-order Plus Dead Time System

4 First-order Plus Dead Time System 

Since dead time is a frequently occurring phenomenon, many systems can be represented by a combination of a first-order system plus dead time (FOPDT) system. In the time domain the FOPDT system equation is Eq. (65). 

For a step-change in input flow rate of magnitude M the system becomes Eqs. (66), the solution to which is Eq. (67). 

If at time to in a system the input variable q o is increased by the amount Aq (, and the output variable instantaneously changes by Ah, then we are dealing with a self-regulating system without time delay. If the output variable follows with de- 

The dead time 9 is the delay in the response after the input step at time to. The time constant t is the time after the output has risen 0.63AH above its original steady value hs after the input step at time to. 

---

The rule developed in Example 27.1 for the sampling rate of a first-order response can be extended to cover a large class of overdamped systems. Figure 16.9 shows the experimental response of an overdamped process to an input step change. The S-shaped response of Figure 16.9 can be approximated by the response of a first-order plus dead time system,... 

Figure 16.8 (a) Process reaction curve (b) its approximation with a first-order plus dead-time system. 

Higher-order systems can be approximated by a first or second-order plus dead-time system for control system design. 

In this case (Cohen and Coon, 1953) it is assirmed that most process systems can be modelled by a first-order plus dead-time model as shown in Eqn. (32.29) ... 

The assumption throughout most of this paper is that the system to be controlled is a first order plus dead time (FOPDT) with a process gain Kp, an overall time constant r and an effective dead time 9, with a transfer function of the form... 

More typically, instead of setpoint changes, the regulatory problem of responding to a system disturbance is encountered in commercial reactors. For this reason, the optimum tuning constants for the PID controller were developed from the IAE ralations for load disturbances. First, however, it is necessary to obtain a process model of the system. Brantley (10) has developed a process identification technique which fits process data to the second order plus dead-time form ... 

The distance over which pneumatic signals can be transmitted is limited by the volume of the tubing and the resistance to flow. The dynamics of pneumatic systems can generally be approximated by a first order lag plus a dead time (Sections 7.S and 7.6). Tubing may be made of copper, aluminium or plastic, and is normally of S mm ID. Pneumatic receivers can be in the form of indicators, recording devices and/or controllers. 

---