Closed Loop Disturbance Gain

Similarly to PRGA, the closed loop disturbance gain (CLDG) is defined as ... 

Compute and plot comparatively the frequency-dependent closed loop disturbance gain (CLDG) and the magnitude of the transfer functions connecting manipulated inputs to outputs. Assess the power and speed of the manipulated variables. 

The previous step gives an indication of the controller tuning the closed-loop disturbance gain (CLDG) should be smaller than the loop transfer function [l+g,v(s), (s)] for each disturbance, where g (s) is the open-loop input/output transfer function, and, (s) the controller model. 

Figure 12.17 Closed loop disturbance gain against loop gain...

Controllability indices, as Closed Loop Disturbance Gain (CLDG) and Performance Relative Gain Array (PRGA) predict in all situations better dynamic properties for the forward heat-integration scheme, compared with the reverse one. This behaviour is verified by closed loop simulation with the full non-linear model. 

Check input constraints by means of closed loop disturbances gain (CLDG). Modify the design if necessary. 

Figure 17.21a shows the closed loop disturbance gain and loop transfer function for the control of I3 with D2 for X13 as disturbance. The loop transfer function with a proportional controller of gain 1 is insufficient to overcome the CLDG element at lower frequencies. To keep I3 between bounds, the loop transfer function should be larger, but the effect cannot be achieved by increasing the scaled controller gain, constrained at maximum 1. The situation is worse in the other alternatives. This analysis has been confirmed by closed loop simulation. The situation can improve if D4 is used instead D2. 

Many of the steady-state tools have been extended to incorporate dynamic information by evaluating the corresponding steady-state measure at multiple frequencies. This can be seen in the dynamic RGA [40-42] and in the closed loop disturbance gain [43]. Jensen, et al. [44] suggests that the best way to present the information from the dynamic RGA analysis is to plot it as an array of polar plots of the magnitude and phase angle for each element. Other measures that extend the use of gain information into the dynamic arena are the interaction quotient [45] and interaction potential [41]. 

Vary the gain to achieve the maximum closed-loop disturbance attenuation. How effective is the controller in rejecting disturbance not already rejected by the natural attenuation of the process ... 

Consider a simple feedback loop (Fig. 7.3a) in which the feedback path consists of elements which approximate to a steady-state gain K (Fig. 7.37). In this instance, the equivalent unity feedback loop is determined by placing 1 IK in the set point input to the main loop and compensating for this by adding an additional factor K in the forward part of the loop prior to the entry of the load disturbance, as in Fig. 7.38. It is easy to confirm that the standard closed loop transfer functions and are the same for the block diagrams in Figs 7.37 and 7.38. 

Suppose that the dead time is increased to td = 0.1. Then the crossover frequency = 17 rad/min and the ultimate gain = 8.56. We notice that the increase in dead time has introduced significant additional phase lag, which reduces the crossover frequency and the maximum allowable gain. In other words, the increase in dead time has made the closed-loop response more sensitive to periodic disturbances and has brought the system closer to the brink of instability. 

Therefore, when the disturbance dn (of the secondary process) changes, the simple feedback controller can use a gain up to 11.88 before the system becomes unstable. Also, given the fact that the overall process is of third order, we expect that the closed-loop response of y(t) to changes in dn will be rather sluggish. 

There is no crossover frequency for the secondary control loop. Therefore, we can use large values for the gain KCiU, which produce a very fast closed-loop response, to compensate for any changes in the disturbance dii arising within the secondary process. 

The comparison of open and closed-loop responses (12.37) and (12.38) reveals that the sensitivity function S gives the reduction of sensitivity to disturbances, achieved by feedback control. It is evident that S =0 and T = 1 are desirable. In this way, the output follows perfectly the setpoint, and the process is not affected by disturbances. Both can be achieved by large controller gain, that is oo. However, large controller gain leads to instability, which sets limits on the achievable closed-loop performance. 

Selection of Control Systems In the operation of control systems stability is required. Operation is stable when continuous cycling will not occur. Instability could be the consequence of the increase in the overall gain of the controller above a maximum value. The overall gain of the controller is the product of gain terms in a closed loop. The role of the time lag may also be considered. Different stabihty aiteria have been elaborated and various rules developed. Integral control and derivative control action can improve the stability of systems added to proportional control. The final performance of a system is affected by the characteristics of the process to be controlled, by the operational characteristics of the controller used, and by the nature of the disturbances to be expected. 

Since the second method does not test the process, the current value of loop gain is unknown until a disturbance Identifies It. Identification must then be carried out, and parameter adjustment made carefully to prevent overcorrection. Identification consists principally of factoring the response curve Into high- and low-frequency components whose ratio represents the dynamic gain of the closed loop. The load-response curve shown in Fig. 6.21 is so separated. 

To further illustrate the influence of controller settings, we consider a simple closed-loop system that consists of a first-order-plus-time-delay model and a PI controller. The simulation results in Fig. 12.1 show the disturbance responses for nine combinations of the controller gain Kc and integral time t/. As Kc increases or t/ decreases, the... 

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