Open-loop gain constant

Maximum value of the open-loop gain constant for the stability of a closed-loop system... 

Value of open-loop gain constant K Applying the magnitude criterion to the above point... 

The Nichols chart shown in Figure 6.26 is a rectangular plot of open-loop phase on the x-axis against open-loop modulus (dB) on the jr-axis. M and N contours are superimposed so that open-loop and closed-loop frequency response characteristics can be evaluated simultaneously. Like the Bode diagram, the effect of increasing the open-loop gain constant K is to move the open-loop frequency response locus in the y-direction. The Nichols chart is one of the most useful tools in frequency domain analysis. 

The schematic and Bode plot for the single-pole method of compensation are given in Figure B-16. At dc it exhibits the full open-loop gain of the op amp, and its gain drops at -20dB/decade from dc. It also has a constant -270 degree phase shift. Any phase shift contributed by the control-to-output characteristic... 

Using MATLAB to design a system, it is possible to superimpose lines of constant ( and ajn on the root locus diagram. It is also possible, using a cursor in the graphics window, to select a point on the locus, and return values for open-loop gain K and closed-loop poles using the command... 

The dominant pole of this temperature control system is also determined by the thermal time constant of the microhotplate, which is approximately 20 ms. The open-loop gain of the differential analog architecture (Aql daa) is given by Eq. (5.8) ... 

Because usually 1/G < 10 the tip closely follows the contour of the constant tunneling current surface of the sample. The higher the open-loop gain G, or the stronger the feedback, the more accurately the tip follows the con-stant-Zr contour. 

This indicates that the oscillation, once set in motion, will be maintained with constant amplitude around the closed-loop for =. % = 0. If, however, the open-loop gain or AR of the system is greater than unity, the amplitude of the sinusoidal signal will increase around the control loop, whilst the phase shift will remain unaffected. Thus the amplitude of the signal will grow indefinitely, i.e. the system will be unstable. 

Certain quantitative measures from linear control theory may help at various steps to assess relationships between the controlled and manipulated variables. These include steady-state process gains, open-loop time constants, singular value decomposition, condition numbers, eigenvalue analysis for stability, etc. These techniques are described in... 

The open-loop gains are determined from steady-state simulation. The responses of the controlled to manipulated variables are expected, in general, to be nonlinear, and therefore the gains are best determined by perturbing each manipulated variable in both directions around the design conditions. For the specified compositions, the design reflux rate is 99.91 kmol/h and the design reboiler duty is 3.491 X 10 kJ/h. Each variable is perturbed by 0.25% while the other is held constant. The results are tabulated below. 

In a normal feedback control loop (Figure 22.2a) the control valve or another of its components may exhibit a nonlinear character. In such a case the gain of the nonlinear component will depend on the current steady state. Suppose that we want to keep the total gain of the overall system constant. From Figure 22.2a we find easily that the open-loop gain is given by... 

In previous considerations of operational amplifier analysis, we have assumed that the device was ideal in the sense that it can respond instantly to any change in response at the inputs. This is equivalent to assuming that the open loop gain is independent of frequency. The characteristic dependence of open loop gain for a real operational amplifier is shown in Fig. 11.8. The roll-off frequency determines the time constant for the potentiostat, Ti, through the relationship... 

Since the controller gain is a factor of the ultimate gain (K = 0.25 K )y the controller gain is proportional to the time constant and inversely proportional to the time delay and the open loop gain. 

If the time delay is much larger than the time constant (Tj To), it can be shown that Equation 8.10s reduces to the ultimate gain being the inverse of the open loop gain. This relationship can also be realized from the amplitude ratio being 1 for a pure time delay. 

To calculate the steady-state gain matrix, open-loop gains can be found by differentiating the model with respect to m, while holding mj (j i ) constant. 

To find the new state feedback gain is a matter of applying Eq. (9-29) and the Ackermann s formula. The hard part is to make an intelligent decision on the choice of closed-loop poles. Following the lead of Example 4.7B, we use root locus plots to help us. With the understanding that we have two open-loop poles at -4 and -5, a reasonable choice of the integral time constant is 1/3 min. With the open-loop zero at -3, the reactor system is always stable, and the dominant closed-loop pole is real and the reactor system will not suffer from excessive oscillation. 

For a first order function with deadtime, the proportional gain, integral and derivative time constants of an ideal PID controller. Can handle dead-time easily and rigorously. The Nyquist criterion allows the use of open-loop functions in Nyquist or Bode plots to analyze the closed-loop problem. The stability criteria have no use for simple first and second order systems with no positive open-loop zeros. 

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