Gain margin proportional controller

If we want to increase the margin, we either have to reduce the value ofKc or increase One possibility is to keep = 1.58 min and repeat the Bode plot calculation to find a new Kc which may provide a gain margin of, say, 2 (6 dB), as in the case of using only the proportional controller. To do so, we first need to find the new ultimate gain using the PI controller ... 

Example 7.2D. Back in the last example with a proportional controller, a gain margin of 1.7 created a system with a very small phase margin. What proportional gain should we use to achieve a phase margin of at least 45° ... 

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 ... 

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. 

For proportional control alone, use a gain margin equal to 2. 

Because a proportional only controller will never reach SP, the quarter decay is determined with respect to the steady state condition. The reciprocal of the coefficient, in this case the reciprocal of 0.5, is known as the gain margin. It is the factor by which the controller gain can be increased before the controller becomes unstable. A proportional only controller tuned according to the Ziegler-Nichols method will therefore have a gain margin of 2. 

The concept of gain and phase margins derived from the Nyquist criterion provides a general relative stability criterion. Frequency response graphical tools such as Bode, Nyquist and Nichols plots can all be used in ensuring that a control system is stable. As in root locus plots, we can only vary one parameter at a time, and the common practice is to vary the proportional gain. 

The reader can easily show that a phase margin of 45° is enough to tune the controller in case 1 and provide the necessary safety factor for absorbing a 50% error in the dead time. The value of the proportional gain Kc for a 45° phase margin is found to be Kc = 5.05. Assume that there is an error in the time constant which has a true value of 0.25 instead of the assumed 0.5. Then the crossover frequency is found from the equation... 

The value of gain for a proportional-only controller that gives 45 of phase margin is Kc = 0.707. 

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