Closedloop servo transfer function

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). 

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. 

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... 

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). 

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. ... 

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. 

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). 

Equation (15.64) gives the effects of setpoint and load changes on tbe controlled variables in the closedloop multivariable environment. The matrix (of order N X N) multiplying the vector of setpoints is the closedloop servo transfer function matrix. The matrix (N x 1) multiplying the load disturbance is the closed-loop regulator transfer function vector. 

A. SCALAR SISO SYSTEMS. Remember in the scalar SISO case we looked at the closedloop servo transfer function G B/ll + GuB). The peak in this curve, the maximum closedloop log modulus L (as shown in Fig. 16.9a), is a measure of the damping coefficient of the system. The higher the peak, the more underdamped die system and the less margin for changes in parameter values. Thus, in SISO systems the peak in the closedloop log modulus curve is a measure of robustness. 

The procedure is to first design the load compensator in the normal way. Then the precompensator is designed using the new closedloop servo transfer function... 

Solving for the closedloop servo transfer function gives... 

So the closedloop servo transfer function 5(5) must be chosen such that Gimc(5) is physically realizable. 

The IMC structure is an alternative way of looking at controller design. The model of the process is clearly indicated in the block diagram. The tuning of the GiMC(i) controller reduces to selecting a reasonable closedloop servo transfer function. 

Clearly the best way to select the closedloop servo transfer function S,j) to make Gimc(.s) physically realizable is... 

There are two basic types of specifications commonly used in the frequency domain. The first type, phase margin and gain margin, specifies how near the openloop GM iu)Gc ia)) polar plot is to the critical (- 1,0) point. The second type, maximum closedloop log modulus, specifies the height of the resonant peak on the log modulus Bode plot of the closedloop servo transfer function. So keep the apples and the oranges straight. We make openloop transfer function plots and look at the (- 1, 0) point. We make closedloop servo transfer function plots and look at the peak in the log modulus curve (indicating an underdamped system). But in both cases we are concerned with closedloop stability. 

Both the openloop and the closedloop frequency response curves can be easily generated on a digital computer by using the complex variables and functions in FORTRAN discussed in Chapter 10 or by using MATLAB software. The frequency response curves for the closedloop servo transfer function can also be fairly easily found graphically by using a Nichols chart. This chart was developed many years ago, before computers were available, and was widely used because it greatly facilitated the conversion of openloop frequency response to closedloop frequency response. 

The numerator and denominator polynomials of the closedloop servo transfer functions are formed using the [numcl,dencl] = cloop(numol,denol, - 1) command. This command converts an openloop transfer function into a closedloop transfer function, assuming negative unity feedback. A block diagram of a unity feedback system is given in Fig. 11.22. 

The magnitudes and phase angles of the closedloop servo transfer functions are... 

In the preceding example we derived the expression for T( ) analytically for a step change in An alternative approach is to use MATLAB to calculate the step response of the closedloop servo transfer function... 

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