Steadystate gains

The ratio of the change in the steadystate value of the output divided by the magnitude of the step change made in the input is called the steadystate gain of the process K,. 

These steadystate gains will be extremely important in our dynamic studies and in controller design. 

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. 

One final point should be made about transfer functions. The steadystate gain Kp for alt the transfer functions derived in the examples was obtained by expressing the transfer function in terms of time constants instead of in terms of poles and zeros. For the general system of Eq. (9.91) this would be... 

The steadystate gain is the ratio of output steadystate perturbation over the input perturbation. 

Thus, for a step change in the input variable of, the steadystate gain is simply found by dividing the steadystate change in the output variable by A5c, , as sketched in Fig. 9.9. 

Instead of rearranging the transfer function to put it into the time-constant form, it is sometimes more convenient to find the steadystate gain by an alternative method that does not require factoring of polynomials. This consists of merely letting s = 0 in the transfer function. 

By definition, steadystate corresponds to the condition that all time derivatives are equal to zero. Since the variable s replaces d/dt in the Laplace domain, letting s go to zero is equivalent to the steadystate gain. 

The final steadystate value of the output will be equal to the steadystate gain since the magnitude of the input was 1. 

For example, the steadystate gain for the transfer function given in Eq. (9.99) is... 

Assume holdups and flow rates are constant. The reaction is an irreversible, first-order consumption of reactant A, The system is isothermal. Solve for the transfer function relating and C. What are the eros and poles of the transfer function What is the steadystate gain ... 

The value of the steadystate gain is unity. Is this reasonable ... 

Calculate the openloop response of this process to a unit step change in its input. What is the steadystate gain of this process ... 

Note that the transfer function for the two-heated-tank process has a steadystate gain that has units of °F/°F. The Gm, ) transfer function has a steady-state gain that has units of °F/Btu/min. 

The openloop transfer function relating steam flow rate to temperature in a feed preheater has been found to consist of a steadystate gain K, and a flrst-order lag with the time constant T. The lag associated with temperature measurement is t . A proportional-only temperature controller is used. 

A process has a positive pole located at (-1-1,0) in the s plane (with time in minutes). The process steadystate gain is 2. An addition lag of 20 seconds exists in the control loop. Sketch root locus plots and calculate controller gains which give a dosedloop damping coeHicient of0.707 when... 

JO. A process has an openloop transfer function that is a first-order lag with a time constant and a steadystate gain K, . If a PI feedback controller is used with a reset time r, sketch root locus plots for the cases where ... 

We are saying that we want the process to respond to a step change in setpoint as a first-order process with a closedloop time constant t,. The steadystate gain between the controlled variable and the setpoint is specified as unity, so there will be no offset. 

The numerical case given is fgr a 20-tray column with 10 trays in the stripping section. A constant relative volatility of 2 1 used. The column steadystate profile is given in Table 12.3, together with the values of coefficients and the transfer functions in terms of log modulus (decibels) and phase angle (degrees) at frequencies from 0 to 10 radians per minute. The values at zero frequency are the steadystate gains of the transfer functions. 

The second,, relates Xi to the controlled variable and is a steadystate gain of 1 and a first-order lag with a time constant of S minutes. 

A first-order lag process with a lime constant of 1 minute and a steadystate gain of 5°F/10 Ib is controlled with a PI feedback controller. 

J2. A process with an openloop transfer function consisting of a steadystate gain, deadtime, and liist-ordcr lag is to be controlled by a PI controller. The deadtime (O) is onc-fiTth the magnitude of the time constant (t). 

As shown in Fig. 14.1, the steadystate gain and deadtime are obtained in the same way as with a first-order model. The damping coefficient can be calculated from the peak overshoot ratio, POR (see Prob. 6.11), using Eq. (14.3). 

If a transfer-function model is desired, approximate transfer functions can be fitted to the experimental curves. First the log modulus Bode plot is used. The low-frequency asymptote gives the steadystate gain. The time constants can be found from the breakpoint frequency and the slope of the high-frequency asymptote. The damping coefficient can be found from the resonant peak. 

The steadystate gain of the transfer function is G(o> or just the ratio of the areas under the input and output curves. 

The other data needed are the steadystate gain and the deadtime. 

The important feature of the ATV method is that it gives transfer function models that fit the frequency-response data very well near the important frequencies of zero (steadystate gains) and the ultimate frequency (which determines closedloop stability). 

The distillation column used in this example separated a binary mixture of propylene and propane. Because of the low relative volatility and large number of trays, the dominant time constant is very large (500 minutes). Despite this large time constant, a sampling period of 9.6 minutes gave poor results. The period had to be reduced to 1,8 minutes to get good identification, both dynamic and steadystate gain. 

A fairly useful stability analysis method is the Niederlinski index. It can be used to eliminate unworkable pairings of variables at an early stage in the design. The settings of the controllers do not have to be known, but it applies only when integral action is used in all loops. It uses only the steadystate gains of the process transfer function matrix. 

If the pairing had been reversed, the steadystate gain matrix would be... 

Example 16.5. Yu and Luyben (Ind. Eng. Chem. Process Des. Dev., 1986, Vol. 25, p. 498) give the following steadystate gain matrices for three alternative choices of manipulated variables reflux and vapor boilup (R — V), distillate and vapor boilup (D — V), and refiux ratio and vapor boilup (RR — V). 

The RGA has the advantage of being easy to calculate and only requires steadystate gain information. Lefs define it, give some examples, show how it is used, and point out its limitations. 

A. DEFINITION. The RGA is a matrix of numbers. The ijth element in the array is called jSy. It is the ratio of the steadystate gain between the ith controlled variable and the jth manipulated variable when all other manipulated variables are constant, divided by the steadystate gain between the same two variables when all other controlled variables are constant. 

For example, suppose we have a 2 x 2 system with the steadystate gains Kp.j... 

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