Closed Loop Responses

Does not introduce instability in the closed-loop response Sensitive to process/model error... 

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. 

Study the simple, open-loop (KC = 0) and closed-loop responses (KC = -1 to 5, TSET = TDIM, and 300 to 350 K) and the resulting yields of B. Confirm the oscillatory behaviour and find appropriate values of KC and TSET to give maximum stable and maximum oscillatory yield. For the open-loop response, show that the stability of operation of the CSTR is dependent on the operating variables by carrying out a series of simulations with varying Tq in the range 300 to 350 K. 

There are other system inputs that can affect our closed-loop response, and we consider them load (or disturbance) variables. In this case, the load variable is the inlet temperature, Tj. Now you may understand why we denote the two transfer functions as Gp and GL. The important point is that input means different things for the process and the closed-loop system. 

In terms of the situation, if we use a PI controller on a slow multi-capacity process, the resulting system response will be even more sluggish. We should use PID control to increase the speed of the closed-loop response (being able to use a higher proportional gain) while maintaining stability and robustness. This comment applies to other cases such as temperature control as well. 

We now apply a different philosophy to controller design. Up until now, we have had a preconceived idea of what a controller should be, and we tune it until we have the desired system response. On the other hand, we can be more proactive we define what our desired closed-loop response should be and design the controller accordingly. The resulting controller is not necessarily a PID controller. This is acceptable with computer based controllers since we are not restricted to off-the-shelf hardware. 

From Eq. (6-20), it is immediately clear that we cannot have an ideal servo response where C/R = 1, which would require an infinite controller gain. Now Eq. (6-21) indicates that C/R cannot be some constant either. To satisfy (6-21), the closed-loop response C/R must be some function of s, meaning that the system cannot respond instantaneously and must have some finite response time. 

That is, no matter what Gp is, we define Gc such that their product is dictated entirely by a function (the RHS) in terms of our desired system response (C/R). For the specific closed-loop response as dictated by Eq. (6-22), we can also rewrite Eq. (6-23) as... 

Model-based) Direct synthesis For a given system, synthesize the controller function according to a specified closed-loop response. The system time constant, xc, is the only tuning parameter. 

Fig. 2.23. The actual tuning relations, based on empirical criteria for the best closed-loop response are given in Table 2.2.

On the other hand, the proposed approach has structure of low pass filter see Eq.(8). Thus, we can expect that the closed-loop response is not sensitive to high frequency signals (as, for example, by noisy measurements or fluctuations in fluids mechanics by agitation). Although Figure 3 depicts the frequency... 

In addition to the visual observations of the dynamic responses, a quantitative measure is needed to provide a better comparison. With such an objective, lAE values were evaluated for each closed-loop response. The PUL option shows the lowest lAE value of 5.607 x 10 , while the value for the Petlyuk column turns out to be 2.35 x 10. Therefore, the results of the test indicate that, for the SISO control of the heaviest component of the ternary mixture, the PUL option provides the best dynamic behavior and improves the performance of the Petlyuk column. Such result is consistent with the prediction provided by the SVD analysis. 

Fig. 5. Closed loop responses for a set point change in the composition of the heavy component...

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. 

Cohen and C00N(27) determined the relationships in Table 7.4 so as to give responses having large decay ratios, minimum offset and minimum area under the closed-loop response curve. 

There is a variety of specifications that can be imposed on the system closed-loop response for a given change in set point. These lead to a number of alternative discrete-time control algorithms—the best known of which are the Deadbeat and Dahlin s algorithms. 

DAHLIN<44) suggested that, in order to avoid the large overshoots and oscillatory behaviour which are characteristic of the deadbeat algorithm, the specification of the system closed-loop response to a step change in set point should be the same as that for a first-order system with dead time. The first-order time constant can then be employed as a design parameter which can be adjusted to give the desired closed-loop response. Hence ... 

Fig. 3. Closed loop response—case study 2 (proposed continuous line, Luyben [14] dotted line).

Closed-loop response to process disturbances and step changes in setpoint is simulated with the model of Kiparissides extended to predict the behavior of downstream reactors. Additionally, a self-optimizing control loop is simulated for conversion control of downstream reactors when the first reactor of the train is operating under closed-loop control with dead-time compensation. 

Acts before the effect of a disturbance has been felt by the system Is good for systems with large time constant or dead time Does not introduce instability in the closed-loop response Requires direct measurement of all possible disturbances Cannot cope with unmeasured disturbances Is sensitive to process/model error... 

Does not require identification and measurement of any disturbance for corrective action Does not require an explicit process model Is possible to design controller to be robust to process/model errors Control action not taken until the effect of the disturbance has been felt by the system Is unsatisfactory for processes with large time constants and frequent disturbances May cause instability in the closed-loop response... 

Pattern recognition self-adaptive controllers exist that do not explicitly require the modeling or estimation of discrete time models. These controllers adjust their tuning based on the evaluation of the system s closed-loop response characteristics (i.e., rise time, overshoot, settling time, loop damp-... 

Figure 7 Closed loop responses of first-order process, (a) P only (set point tracking), (b) P only (disturbance rejection), (c) stable PI or PID (disturbance rejection) and (d) unstable PI or PID (disturbance rejection).

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