Feedback controllers closed loop responses

The tuning of the controller in the feedback loop can be theoretically performed independent of the feedforward loop (i.e., the feedforward loop does not introduce instability in the closed-loop response). For more information on feedforward/feedback control appications and design of such controllers, refer to the general references. 

Forcing function is a term given to any disturbance which is externally applied to a system. A number of simple functions are of considerable use in both the theoretical and experimental analysis of control systems and their components. Note that the response to a forcing function of a system or component without feedback is called the open-loop response. This should not be confused with the term open-loop control which is frequently used to describe feed-forward control. The response of a system incorporating feedback is referred to as the closed-loop response. Only three of the more useful forcing functions will be described here. 

A related approach which has been used successfully in industrial applications occurs in discrete-time control. Both Dahlin (43) and Higham (44) have developed a digital control algorithm which in essence specifies the closed loop response to be first order plus dead time. The effective time constant of the closed loop response is a tuning parameter. If z-transforms are used in place of s-transforms in equation (11), we arrive at a digital feedback controller which includes dead time compensation. This dead time predictor, however, is sensitive to errors in the assumed dead time. Note that in the digital approach the closed loop response is explicitly specified, which removes some of the uncertainties occurring in the traditional root locus technique. 

Sensor readings shonld nsnally be hltered to reduce the influence of sensor noise on feedback control performance. Filtering, however, adds lag to the closed-loop response. In certain cases, tnning a hlter on a sensor can involve balancing the benefits of reducing the noise against the... 

The system in Figure 13.1a is known as open loop, in contrast to the feedback-controlled system of Figure 13.1b, which is called closed loop. Also, when the value of d or m changes, the response of the first is called open-loop response while that of the second is the closed-loop response. The origin of the term closed-loop is evident from Figure 13.1b. 

In all of the situations noted above, a conventional feedback controller would provide quite unsatisfactory closed-loop response, for the following reasons ... 

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. 

In the feedback control loop of Figure 29.9 we have omitted the dynamics of the measuring sensor and final control element. Thus they are absent from the closed-loop response of eq. (29.21). Consider the loop of Figure 29.10, with the sensor and final control element included. Following the same procedure as above, it is easy to show that the sampled-value, closed-loop response of the loop is given by... 

Consider a first-order process with a PI feedback controller. The closed-loop response to set point changes is given by... 

Consider the generalized feedback control system shown in Figure 14.1. The closed-loop response for such system is given by eq. (14.5) ... 

Consider two different feedback control systems producing the two closed-loop responses shown in Figure 16.2. Response A has reached the desired level of operation faster than response B. If our criterion for the design of the controller had been... 

The Bode stability criterion indicates how we can establish a rational method for tuning the feedback controllers in order to avoid unstable behavior by the closed-loop response of a process. 

IV.23 Each of the following systems is feedback controlled with a proportional controller. Find the range of values of the proportional gain Kc that produce stable (if it is possible) closed-loop responses. Also, identify the characteristic equations. Assume that Gm = G/= 1. 

The closed-loop response under feedback control becomes ... 

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. 

The basic idea of IMC is to use a process model and to relate the controller settings to the model parameters in such a way that the selection of the specified closed-loop response yields a physically realizable feedback controller [8,17]. IMC is advantageous because it can be adjusted to balance controller performance with control system robustness (when either modeling errors or changes in process dynamics occur). Clearly, the effectiveness of IMC depends on the availability of a reliable model for the dryer. 

A three point control configuration based on a proportional-integral controller with dynamic estimation of unknown disturbances was implemented in a Petlyuk column. The proposed controller comprises three feedback terms proportional, integral and quadratic actions. The first two terms act in a similar manner as the classical PI control law, while the quadratic term (double integral action) accounts for the dynamic estimation of unknown disturbances. Comparison with the classical PI control law was carried out to analyze the performance of the proposed controller in face to unknown feed disturbances and set point changes. The results show that the closed-loop response of the Petlyuk column is significantly improved with the proposed controller. 

The nonlinear simulation was used to illustrate the closed-loop response of the controlled variable X2 following a 30 percent increase in feed composition. The results are shown in Figure 21.4b with the feedback-only dual and PID algorithms. Control is immensely improved with the feedforward action. The slight deviation in X2 with feedforward control is due to inaccuracies in the linear model and the long sampling time relative to the process dead time. The... 

Consider the blending system of Section 15.3, but now assume that a pneumatic control valve and an I/P transducer are used. A feedforward-feedback control system is to be designed to reduce the effect of disturbances in feed composition xi on the controlled variable, product composition X. Inlet flow rate 1V2 can be manipulated. Using the information given below, design the following control systems and compare the closed-loop responses for a +0.2 step change in xi. 

Figure 15,13 Comparison of closed-loop responses (a) feedforward controllers with and without dynamic compensation (b) feedback control and feedforward-feedback control.

The closed-loop responses to a step change in x fron 0.2 to 0.4 are shown in Fig. 15.13. The set point is the nominal value, Xsp = 0.34. The static feedforward controller for cases (a) and (b) are equivalent and thus produce identical responses. The comparison in part (a) of Fig. 15.1 shows that the dynamic feedforward controller is superior to the static feedforward controller, because it provides a better approximation to the ideal feedforward controller of Eq. 15-33. The PI controller in part (b) of Fig. 15.13 produces a larger maximum deviation than the dynamic feedforward controller. The combined feedforward-feedback control system of part (d) results in better performance than the PI controller, because it has a much smaller maximum deviation and lAE value. The peak in the response at approximately t = 13 min in Fig. 15.13b is a consequence of the x measurement time delay. 

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