Maximum closedloop log modulus

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 ... 

A commonly used maximum closedloop log modulus specification is 4 2 dB. The controller parameters are adjusted to give a maximum peak in the closedloop servo log modulus curve of -1-2 dB. This corresponds to a magnitude ratio of 1.3 and is approximately equivalent to an underdamped system with a damping coefficient of 0.4,... 

Maximum closedloop log modulus. We have already designed in Sec. 13.2.3 a proportional controller that gave an of +2 dB. Figure 13.1 If gives a... 

The Bode plot of is given in Fig. 13.20 for D = 0.5. The ultimate gain is 3.9 (11.6 dB), and the ultimate frequency is 3.7 radians per minute. The ZN controller settings for P and PI controllers and the corresponding phase and gain margins and log moduli are shown in Table 13.2 for several values of deadtime D. Also shown are the values for a proportional controller that give +2-dB maximum closedloop log modulus. 

If a single proportional controller is used to control by manipulating M, deter-mine the gain that gives a phase margin of 4S degrees. What is the maximum closedloop log modulus when this gain is used ... 

J0. A process has an openloop transfer function that contains a positive pole at -H 1/t, a negative pole at — 10/t and a gain of unity. If a proportional-only controller is used, find the two values of controller gain that give a maximum closedloop log modulus of -1-2 decibels. 

If a PI controller is used, find the smallest value of the ratio of the reset time T/ to, the process time constant for which a maximum closedloop log modulus of + 2 dedbds is attainable. 

If a proportional controller is used in the three-isothermal CSTR process, a controller gain of 22.6 gives a phase margin of 45°. A gain of 20 gives a maximum closedloop log modulus of +2 dB with a closedloop resonant frequency of 1.1 radian per minute. 

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 basic idea proposed by Skogestad and Morari is to design the controller so that the closedloop resonant frequency of the system occurs at a frequency that is lower than the region where the uncertainty becomes significant. For example, suppose Kp = Tp = 1 in Eq. (16.44). If a proportional controller is used and a tuning criterion of - - 2 dB maximum closedloop log modulus is assumed, the value of the controller gains and closedloop resonant frequencies for different deadtimes are hsted below. 

The F factor is varied until L ls equal to 2N, where N is the order of the system. For N = 1, the SISO case, we get the familiar -f2 dB maximum closedloop log modulus criterion. For a 2 x 2 system, a +4 dB value of is used for a 3 x 3, -1-6 dB and so forth. This empirically determined criterion has been tested on a large number of cases and gives reasonable performance, which is a little on the conservative side. 

Note that this is smaller than the ultimate gain for the process with no deadtime. The controller gain that gives + 2 dB maximum closedloop log modulus is = 1.36. 

In the frequency domain, the conventional criteria of phase margin, gain margin, or maximum closedloop log modulus are used. The shape of the or curve is modified by changing The simplest form of a 0,, sampled-data controller is... 

Unfortunately the sampled-data controller is really not better than the continuous. The controller gain that gives 4- 2 dB maximum closedloop log modulus for the continuous controller is 1.95. The corresponding gain for a sampled-data controller with a sampling period of 1.1 is only 1.73. In addition, the sampled-data controller will not detect load disturbances as quickly as the continuous. So the performance of the sampled-data controller is not as good as the continuous. 

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. 

The most useful frequency-domain specification is the maximum closedloop log modulus. The phase margin and gain margin specifications can sometimes give poor results when the shape of the frequency response curve is unusual. 

If a proportional-only controller is used, what value ofcofitroller gain will give a maximum closedloop log modulus of +2 dB ... 

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