Lead compensation

Design a cascade lead compensator that will ensure stability and provide a phase margin of at least 30°, a bandwidth greater than 5rad/s and a peak closed-loop modulus Mp of less than 6dB. 

The open-loop transfer function is third-order type 2, and is unstable for all values of open-loop gain K, as can be seen from the Nichols chart in Figure 6.33. From Figure 6.33 it can be seen that the zero modulus crossover occurs at a frequency of 1.9 rad/s, with a phase margin of —21°. A lead compensator should therefore have its maximum phase advance 0m at this frequency. Flowever, inserting the lead compensator in the loop will change (increase) the modulus crossover frequency. 

Fig. 6.35 Bode gain and phase for lead compensator, design one.

Fig. 6.36 Closed-loop frequency response for lead compensator one.

Fig. 6.37 Open-loop bode gain and phase for design two lead compensator.

Fig. 6.39 Closed-loop frequency response for both lead compensator designs.

A lead compensator, see case study Example 6.6, and equation (6.113) has a transfer function of... 

Nichols Chart for Case Study Example 6.6 %Lead Compensator Design One, Figure 6.34 clf... 

The nice feature of the phase-lead compensator, and for that matter a real PD controller, is that it limits the high frequency magnitude. In contrast, an ideal PD controller has no upper limit and would amplify high frequency input noises much more significantly. 

Normal commercial PID controllers are generally constructed by adding a lead compensator (Section 7.12.2) as the derivative mode to a PI controller. This type of derivative module typically has the transfer function ... 

The lead compensator contributes phase advance to the system and thus increases the overall system stability (Section 7.10.4). The degree of phase advance provided is a function of frequency. At the same time this type of compensator increases the overall system amplitude ratio, which has the effect of reducing the the stability of the system. However, the major contribution of phase advance occurs at those frequencies where the open-loop polar plot is adjacent to the (-1,0) point on the complex plane. The increase in amplitude ratio takes place at lower frequencies and, consequently, the effect of this is much less significant. As the ratio of r,/r2 is increased, the maximum phase advance supplied by the lead compensator also increases, i.e. the greater is the stabilising effect of the compensating element011. 

Curve a in Fig. 7.62 shows the open-loop polar plot for the heat exchanger system described in Example 7.6. with Kc = 1.8 and t,= 2.5 (see also Example 7.8 and Fig. 7.55). Clearly this indicates an unstable system (Section 7.10.5). If a lead compensator with r, = 1 min and r2 = 0.1 min (r,/r2 = 10) is inserted into the loop, as shown in Fig. 7.63, then the system becomes stable (curve b in Fig. 7.62) due to the additional phase lead supplied by the compensator. (Using these values of r, and r2, Kc can now be increased by almost a factor of ten before the system becomes unstable). 

The properties of the lead compensator must be considered in order to select suitable values for rt and tj. The AR and phase shift of such a compensator are (from equations 7.104, 7.105 and 7.149) ... 

Fig. 7.56). Determine the values of r, and r2 for a series lead compensator such that the compensated system will exhibit a gain margin of 3.8.

The output of the element represented by equation 7.155 lags the input. However, the destabilising effect of this additional lag is more than offset by an associated decrease in amplitude ratio. This decrease is more pronounced as the difference between r, and tj is increased. Lag compensators can be designed to produce different total open-loop stability specifications (e.g. in terms of allowable gain margin, phase margin, etc.) in a manner similar to that for lead compensators. 

Lag-Lead Compensation The transfer function of a lag-lead element is ... 

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