First-order lag system

The Bode diagram, given in Figure 6.10, is the mirror image, about the frequeney axis, of the first-order lag system. Note that the transfer funetion given in equation (6.35) is also that of a PD eontroller. 

Figure 10.5 Effect of (a) time constant and (b) static gain, in the response of first-order lag systems.

Find the dynamic response of a first-order lag system with time constant tp = 0.5 and static gain Kp = 1 to (a) a unit impulse input change, (b) a unit pulse input change of duration S, and (c) a sinusoidal input change, sin 0.51. Determine the behavior of the output after long time (as / - oo) for each of the input changes above. 

Figure 29.6 Pulse transfer functions for (a) pure integrator (b) first-order lag system.

Exponential transfer lag system (see First-order lag system)... 

Eq. (10.13) demonstrates clearly that this is a first-order lag system. 

Consider a closed vessel with air flowing in it. Is this a pure capacitive or a first-order lag system Answer the same question if the vessel is also supplied with an exit for the air. 

First-Order Lag (Time Constant Element) Next consider the system to be the tank itself. A dynamic mass balance on the tank gives ... 

Higher-Order Lags If a process is described by a series of n first-order lags, the overall system response becomes proportionally slower with each lag added. The special case of a series of n first-order lags with equal time constants has a transfer function given by ... 

Well-mixed tank systems (Fig. 2.18) are characterised by a first-order lag response. 

For a better fit of the system response, the method of Oldenbourg and Sartorius, as described in Douglas (1972), using a combination of two first-order lags plus a time delay, can be used. The method is illustrated in Fig. 2.24. and applies for the case... 

On the Bode plot, the comer frequencies are, in increasing order, l/xp, Zq, and p0. The frequency asymptotes meeting at co = l/xp and p0 are those of a first-order lag. The frequency asymptotes meeting at co = z0 are those of a first-order lead. The largest phase lag of the system is -90° at very high frequencies. The system is always stable as displayed by the root locus plot. 

This particular type of transfer function is called a first-order lag. It tells us how the input affects the output C/, both dynamically and at steadystate. The form of the transfer function (polynomial of degree one in the denominator, i.e., one pole), and the numerical values of the parameters (steadystate gain and time constant) give a complete picture of the system in a very compact and usable form. The transfer function is property of the system only and is applicable for any input. 

If C > 1, show that the system transfer function has two first-order lags with time constants r,i and. Express these time constants in terms of Tj, and . 

Example 10.6. Let us start with the simplest of all processes, a first-order lag. We will choose a proportional controller. The system and cooiroller transfer functions are... 

The systems explored above illustrate a very important point about the control of openloop unstable systems the control of these systems becomes more difficult as the order of the system is increased and as the magnitudes of the first-order lags increase. 

Mathematically, inverse response can be represented by a system that has a transfer function with a positive zero, a zero in the RHP. Consider the system sketched in Fig. ll.lOn. There are two parallel first-order lags with gains of opposite sign. The transfer function for the overall system is... 

From the three eases above we can conclude that, for this third-order system with three equal first-order lags, the 4-2-dB specification is the most conservative, the 45° PM is next, and the 2 GM gives the controller gain that is closest to instability. 

Example 19.4. Adding a deadtime to the first-order lag process that is equal to two sampling periods gives a third-order system in the z plane. 

Example 19.6. The chromatographic system studied in Example 18.9 had a first-order lag openloop process transfer function and a deadtime of one sampling period. The closedloop characteristic equation was [see Eq. (18.100)]... 

Example 19.7. The first-order lag process, zero-order hold, and proportional sampled-data controller from Example 19.1 gave an openloop system transfer function... 

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. 

If the unsteady-state behaviour of a system is described by a first order differential equation, it is termed a first-order system. Other descriptions frequently used are first-order lag and single capacity system. Similarly, a component described by a second-order differential equation is termed a second-order system, and so on. 

There are distinct similarities between second order systems and two first-order systems in series. However, in the latter case, it is possible physically to separate the two lags involved. This is not so with a true second order system and the mathematical representation of the latter always contains an acceleration term (i.e. a second-order differential of displacement with respect to time). A second-order transfer function can be separated theoretically into two first-order lags having the same time constant by factorising the denominator of the transfer function e.g. from equation 7.52, for a system with unit steady-state gain ... 

Model representations in Laplace transform form are mainly used in control theory. This approach is limited to linear differential equation systems or their linearized approximations and is achieved by a combination of first-order lag function and time delays. This limitation together with additional complications of modelling procedures are the main reasons for not using this method here. Specialized books in control theory as mentioned above use this approach and are available to the interested reader. 

A first-order lag process is self-regulating. Unlike a purely capacitive process, it reaches a new steady state. In terms of the tank system in the Example 10.1, when the inlet flow rate increases by unit step, the liquid level goes up. As the liquid level goes up, the hydrostatic pressure increases, which in turn increases the flow rate F0 of the effluent stream [see eq. (10.5)]. This action works toward the restoration of an equilibrium state (steady state). 

What is a first-order system, and how do you derive the transfer functions of a first-order lag or of a purely capacitive process ... 

Dividers, 428 (see also Ratio control) Drum boiler, 105, 216-17, 412-13 Drying control, 456 Duhem s rule, 97 Dynamic analysis, 51 qualitative characteristics, 168-72 Dynamic behavior of various systems dead time, 214-16 definition, 51 first-order lag, 179-83 higher-order, 212-14 inverse response systems, 216-20 pure capacitive, 178-79 second-order, 187-93... 

Figure 10.1 Systems with capacity for mass storage (a) first-order lag (b) pure capacitive.

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