Transfer function closedloop

In this chapter we will demonstrate the signiScant computational and nota-tional advantages of LaplaTce transforms. The techniques involve finding the transfer function of the openloop process, specifying the desired performance of the closedloop system (process plus controller) and finding the feedback controller transfer function that is.required to do the job. 

Closedloop Characteristic Equation and Closedloop Transfer Functions... 

Equation 10.5 gives the transfer functions describing the closedloop system, so these are closedloop transfer functions. The two inputs are the load L, and the setpoint The controlled variable is Note that the denominators of both of these closedloop transfer functions are identical. 

Example 10.1. The closedloop transfer functions for the two-heated-lank process can be calculated from the openloop process transfer functions and the feedback controller transfer function. We will choose a proportional controller, so = K,. Note that the dimensions of the gain of the controller are mA/mA, that is, the gain is dimensionless. The controller looks at a milliampere signal (PM) and puts out a milliampere signal (CO). 

Notice that the denominators of all of these closedloop transfer functions are identical. Notice also that the steadystatc gain of the closedloop servo transfer function PM/SP is not unity i.e., there is a steadystate offset. This is because of the proportional controller. We can calculate the PM/SP ratio at steadystate by letting s go to zero in Eq. (10.8). 

Since the characteristic equation of any system (openloop or closedloop) is the denominator of the transfer function describing it, the closedloop characteristic equation for this system is... 

This equation shows that closedloop dynamics depend on the process openloop transfer functions (G, Gv, and Gj) and on the feedback controller transfer function (fl). Equation (10.10) applies for simple single-input-single-output systems. We will derive closedloop characteristic equations for other systems in later chapters. 

The first closedloop transfer function in Eq. (10.5) relates the controlled variable to the load variable. It is called the closedloop regulator transfer function. The second closedloop transfer function in Eq. (10.5) relates the controlled variable to the setpoint. It is called the closedloop servo transfer function. 

It is useful to consider the ideal situation. If we could design an ideal controller without any regard for physical realizability, what would the ideal elosed-loop regular and servo transfer functions be Clearly, we would wish a load disturbance to have no effect on the controlled variable. So the ideal closedloop regulator transfer function is zero. For setpoint changes, we would like the controlled variable to track the setpoint perfectly at all times. So the ideal servo transfer function is unity. 

This stability requirement applies to any system, openloop or closedloop. The stability of an openloop process depends upon the location of the poles of its openloop transfer function. The stability of a closedloop process depends upon the location of the poles of its closedloop transfer function. These closedloop poles will naturally be different from the openloop poles. 

Find the ultimate gain and period of a closedloop system with a proportional controller and an openloop transfer function ... 

Derive a dynamic mathematical model of the flooded-condenser system. Calculate the transfer function relating steam flow rate to condensate flow rate. Using a PI controller with tj = 0.1 minute, calculate the closedloop time constant of the steam flow control loop when a closedloop damping coefTident of 0.3 is used. Compare this with the result found in (u). 

Now we solve for the closedloop transfer function for the primary loop with the secondary loop on automatic. Figure 11.3c shows the simplified block diagrana. By inspection we can see that the closedloop characteristic equation is... 

The addition of the feedforward controller has no effect on the closedloop stability of the system for linear systems. The denominators of the closedloop transfer functions are unchanged. 

Keep in mind that the positive zero does not make the system openloop unstable. Stability depends on the poles of the transfer function, not on the zeros. Positive zeros in a system do, however, affect closedloop stability as the example below illustrates. 

The root locus curves are shown in Fig. 11.10c. The loci start at the poles of the openloop transfer function . s = — 1 and 5 = — Since the loci must end at the zeros of the openloop transfer function (.t — + ) the curves swing over into the RHP. Therefore the system is closedloop unstable for gains greater than 2. 

Now, knowing the process model and having specified the desired closedloop servo transfer funclion, we can solve for the feedback controller transfer function. We define the closedloop servo transfer function as. ... 

Equation (11.65) is a general solution for any process and for any desired closedloop servo transfer function. Plugging in the values for and for the specific example gives... 

This last case illustrates that the desired closedloop relationship cannot be chosen arbitrarily. You cannot moke a jumbo jet behave like a jet fighter We must select the desired response such that the controller b physically realizable. In this case all we need to do is modiiy the specified closedloop servo transfer function to include the dcadtime. 

Specifying the original closedloop servo transfer function [Eq. (11.63) and solving for the feedback controller using Eq. (11.65) gives... 

This type of controller design has been around for many years. The pole-placcmcnt methods that are used in aerospace systems use the same basic idea the controller is designed so as to position the poles of the closedloop transfer function at the desired location in the s plane. This is exactly what we do when we specify the closedloop time constant in Eq. (11.63). 

The basic idea of IMC is to use a model of the process openloop transfer function in such a way that the selection of the specified closedloop response yields a physically realizable feedback controller. 

So if we cannot attain perfect control, what do we do From the IMC perspective we simply break up the controller transfer function C( ) into two parts. The first part is the inverse of. The second part, which Morari calls a filter, is chosen to make the total physically leahzable. As we will show below, this second part turns out to be the closedloop servo transfer function that we defined as S(,j in Eq. (11.64). 

Derive the closedloop transfer function between and What is the closedloop characteristic equation ... 

G. SECOND OHDER UNDERDAMPED SYSTEM. This is probably the most important transfer function that we need to translate into the frequency domain. Since we often design for a desired closedloop damping coeflident, we need to know what the Nyquist plot of such a system looks like. 

Note the very unique shape of the log modulus curves in Fig. 12.19. The lower the damping coefficient, the higher the peak in the L curve. A damping coefficient of about 0.4 gives a peak of about +2 dB, We will use this property extensively in our tuning of feedback controllers. We will adjust the controller gain to give a maximum peak of +2 dB in the log modulus curve for the closedloop servo transfer function X/X. ... 

Keep in mind that we are talking about closedloop stability and that we are studying it by making frequency-response plots of the total openlaop system transfer function. We are also considering openlocp stable systems most of the time. We will show how to deal with openloop unstable processes in Sec. 13.4. 

The maximum closedloop log modulus does not have these pioblems since it measures directly the closeness of the G B curve to the (—1,0) point at all frequencies. The closedloop log modulus refers to the closedloop servo transfer function ... 

All the Nyquist, Bode, and Nichols plots discussed in previous sections have been for openloop system transfer functions B(j ). Frequency-response plots can be made for any type of system, openloop or closedloop. The two closedloop transfer functions that we derived in Chap. 10 show how the output is affected in a closedloop system by a setpoint input and by a load. Equation (13.28) gives the closedloop servo transfer function. Equation (13.29) gives the closedloop load transfer function. 

Typical log modulus Bode plots of these two closedloop transfer functions are shown in Fig. 13.10a. If it were possible to achieve perfect or ideal control, the two ideal closedloop transfer functions would be... 

Figure 13.20 shows the closedloop servo transfer function Bode plots for P and PI controllers with the ZN settings for a deadtime of 0.5 min. The effect of... 

Notice in Fig. 13.20 that the curve for the P controller does not approach 0 dB at low frequencies. This shows that there is a steadystate offset with a proportional controller. The curve for the PI controller does go to 0 dB at low frequencies because the integrator drives the closedloop servo transfer function to unity (i.e., no offset). 

First of all, we know immediately that the openloop system transfer function has one pole (at s = -I- 1/ip) in the RHP, Therefore the closedloop... 

A process has and openloop transfer functions that are first-order lags and gains Tm. tj, and X. Assume -r is twice Tj,. Sketch the log modulus Bode plot for the closedloop load transfer function when ... 

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